Nick Hu https://forest.nickx.hu/index/ index /index/ The root of my forest Welcome to my forest! Here are my work-in-progress public notes. My thesis is also written in Forester. I am currently running a reading group on double categories. journal References Context Backlinks Related Nick Hu 2023 12 3 https://forest.nickx.hu/coherent-inverses/ coherent-inverses /coherent-inverses/ Coherent inverses in higher-categorical string diagrams project Nick Hu2023123https://forest.nickx.hu/coherent-inverses-abstract/coherent-inverses-abstract/coherent-inverses-abstract/abstract The proof assistant homotopy.io serves as a combinatorial model for finitely presented globular higher categories in the form of a computer implementation, based on the theory of zigzag categories, presented as a calculus of -dimensional string diagrams. However, the theory lacks a notion of inverse, which would require the existence of cancellation moves along with an infinite collection of higher-dimensional coherence data. We generalise the theory to support invertibility by equipping algebraic generators with ‘framing’ data, allowing for a natural representation of the inverse. Using this ‘framed zigzags’ approach, we show that the required cancellation moves and coherence data arise uniformly via a colimit mechanism. We use tools from enriched category theory to justify correctness of our constructions, and exhibit an enhanced version of our proof assistant that implements our generalised theory. We give an account of the design and implementation of homotopy.io from a programming perspective, and demonstrate how it can be used to formalise higher-categorical arguments using this new notion of coherent inverse, presented as -dimensional string diagrams, for the first time. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-acknowledgements/coherent-inverses-acknowledgements/coherent-inverses-acknowledgements/acknowledgements Firstly, I would like to thank my supervisors Sam Staton and Jamie Vicary for years of generous support and guidance. Jamie has been a constant source of both professional and personal advice, and I am especially grateful for his patience and encouragement to pursue my own ideas independently. I am grateful to Aleks Kissinger, Dan Marsden, and Amar Hadzihasanovic for volunteering their time in various examinations of this thesis. Over the years, I am privileged to have had the opportunity to meet and befriend many talented people across the world. I am lucky to have been active in two vibrant research groups: the Quantum Group at the University of Oxford, and the Theory Group at the University of Cambridge. In no particular order, I would like to thank my friends with whom I shared fond memories, which will surely come to define the experience of my PhD, in Oxford, Cambridge, and various conferences: Vincent Wang, Lukas Heidemann, Cole Comfort, Nihil Shah, Amin Karamlou, Alex Rice, Calin Tataru, Ioannis Markakis, Chiara Sarti, Manuel Araújo, Meven Lennon-Bertrand, Thibaut Benjamin, Wilf Offord, and Jesse Sigal. I wrote half of this thesis far away from home, on a series of academic visits in Australia and Japan in the summer of 2024, partly thanks to the support of JS Lemay, Masahito Hasegawa, and Ichiro Hasuo. During my travels, I learned a lot as a category theorist, and I would like to thank the following additional people for their hospitality in spending time with me, whether that was in teaching me category theory, showing me around, or procrastinating from thesis writing together: Richard Garner, Eli Hazel, Nicola Di Vittorio, Vincent Moreau, Jean Zablocki, Hayato Nasu, and Keisuke Hoshino. Lastly, and most importantly, I would like to thank my partner Amy Chang for her unwavering love and selflessness — without her, I would not have had any chance of making it this far — and Sesame the Cat for being a constant distraction in the best of ways. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-introduction/coherent-inverses-introduction/coherent-inverses-introduction/Introductionchapter This thesis studies the theory and implementation of the proof assistant homotopy.io for finitely presented globular -categories, viewed through the lens of -dimensional string diagrams (also known as manifold diagrams, generalising surface diagrams where and string diagrams where ). homotopy.io is based on the semistrict model of higher categories called associative -categories (ANCs), which is conjecturally equivalent to weak models of higher categories. We focus on coherent inverses within homotopy.io, a new feature that enables more expressivity, allowing one to study not just -categories, but also -groupoids, and necessitating new theory development. Additionally, we give an account of the implementation of homotopy.io, and explore its applications in deriving new, previously inaccessible, proofs of higher-dimensional phenomena arising from coherent inverses in higher categories. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-higher-categories/coherent-inverses-higher-categories/coherent-inverses-higher-categories/What is a ‘higher category’?section Unfortunately, even the term ‘higher category’ as it appears in the literature is ambiguous, in more than one way. A precise lexicon for one axis of this ambiguity is the notion of -category, where ; this refers to a mathematical structure (like a category) in which there are dimensions of composition, but every -morphism, for , is invertible. Often, the term ‘-category’ is taken to mean a notion of higher category that has infinitely many dimensions of composition, but for which each -morphism for is invertible. Such structures are more precisely called -categories, and are designed to model a more homotopy-invariant version of mathematics, with -groupoids providing a foundation for studying ‘spaces up to homotopy’ via the homotopy hypothesis. Another frequently used term is ‘-category’, which can be taken to be a shorthand for -category; this is closer to what we mean by ‘higher category’ in this thesis, and by ‘-category’ we mean -category in this sense. The second axis of ambiguity is that of strictness. The basic idea of ‘strict vs weak’ is whether equations regarding composition must hold up to equality, or if instead it is permissible for them to hold up to some good notion of ‘equivalence’. A relatively simple example of this is strict vs weak 2-categories (which are commonly referred to as simply ‘2-categories’ and ‘bicategories’ respectively). In a bicategory, composition and unit laws are allowed to hold up to invertible 2-morphism, as opposed to strict 2-categories where such laws hold on the nose up to equality. In dimensions 2]]>, it is known that weak notions of -category are properly more expressive than strict notions of -category, meaning that they model more mathematical structures. For instance, for some suitable notion of equivalence, not every tricategory (a weak 3-category) is equivalent to some strict 3-category (however, the converse is trivially true). While models for weak notions of 3-category have been described as Gray categories, there is no generally accepted model of weak -category for arbitrary ; however, it is widely accepted that, just as a category is a mathematical structure that has fundamental support for a single ‘composition’ operation, a weak -category should be a mathematical structure that has different dimensions of composition that should cohere nicely together in some way. Fully weak models of -categories are difficult to work with, as their explicit description is intractably large for even 2]]>, so in practice it is convenient to instead use a so-called semistrict model. Such a model has some of its structure holding strictly, e.g. the unit law for composition holds up to equality, but crucially retains some weak aspects to preserve equivalence with the fully weak notion. In this sense, a Gray category can be considered a semistrict model of tricategories, which has strict associativity and unit laws, but weak interchange. The idea of ‘Gray semistrictness’ is the conjecture that this extends past tricategories to arbitrary weak -categories. In a nutshell, this is the flavour of ANCs — strict associativity and unit laws for composition in all -dimensions, but with weak interchange. One of the tenets of category theory, which is historically rooted in algebraic topology, is that there is a deep connection between the algebraic and the geometric; algebraic structures are usually mathematically tractable to both define and manipulate, being the typical foundation for computer-based mathematics systems, but topological structures are, although wilder in this regard, in some sense more intuitive and provide a more semanticlly meaningful form of mathematical argumentation. Any concept arising in one aspect should have meaning in the other, especially in the study of higher categories where the space-like aspects enter the fore. The other motivating principle for ANCs is that instead of defining a model of semistrict -category in an algebraic vein, as is done for Gray categories, which is inextricably difficult for arbitrary , it is more fruitful to approach it from the other end geometrically — via higher-dimensional string diagrams. A simple extension of the usual (2D) string diagrammatic calculus with coloured regions is a mathematical language for strict 2-categories, where 2-morphisms are represented by string diagrams [§ 8, a-survey-of-graphical-languages-for-monoidal-categories]. Implicit in this construction is the idea that every bicategory can be strictified ​ , so such a string diagrammatic calculus completely captures the language of 2-categories. Indeed, the idea that string diagrams may be juxtaposed to represent composites is essentially a realisation of the strictness of associativity, while the length-invariance of wires in a string diagram realises the strictness of unitality. Weak interchange can be represented in a string-diagrammatic way by replacing equations of string diagrams with ‘homotopies’: i.e., a string diagram of higher dimension that admits the left side of the equation as a face and the right side as the opposing face. In other words, mere equality is replaced by a proof-relevant equality. In this sense, ANCs stem from the idea that formally distilling this higher-dimensional string diagram calculus is easier, and from this we can reverse-engineer a model of Gray semistrict higher -category. Finally, the last adjective ‘globular’, which we use to describe our notion of higher category, refers to the shape of the cells from which higher morphisms are constructed, in this case as ‘globes’, which are discs generalised to arbitrary dimension. For example, in the definition of a natural transformation between two functors, one must assume that both functors have matching domain and codomain (hence the usual diagram that one draws has the shape of a disc, a 2-globe). Just as space can be decomposed into a variety of shapes, there are different combinatorial notions of higher category based on choosing different fundamental shapes, and moreover these may model globular higher categories. Quasicategories, a popular model of (globular) -category, are based on choosing simplices as the primitive shapes. Other choices of shapes include cubes, opetopes, and diagrammatic sets, all of which would replace the role of ANCs in this thesis to model globular higher categories. Globes are fundamental to higher categories (based on iterated enrichment) due to the previous observation about natural transformations, which extends to their higher-dimensional analogues (so-called -transfors): modifications, pertubations, etc. By non-globular higher category, we mean a notion of ‘higher category’ which is not based on iterated enrichment, e.g. the -fold or ‘multiple’ categories obtained by iterated internalisation. While there do exist string diagrammatic calculi for some forms of non-globular higher categories based on other shapes (e.g. for double categories), those based on globular higher categories are by far the most well-studied (as these generalise most traditional string diagrammatic calculi of various types of monoidal category), so in this thesis every -category is implicitly globular, as opposed to -fold or ‘multiple’ categories. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-combinatorial-foundation/coherent-inverses-combinatorial-foundation/coherent-inverses-combinatorial-foundation/Combinatorial foundationsection Although we have established our appropriate notion of ‘higher category’, it is still not clear in which mathematical arena our work takes place. Because our work focuses on the proof assistant homotopy.io, which in some sense implements the theory of ANCs, we seek to have a combinatorial description of this theory along with a categorical semantics of our implementation. This foundation is given by zigzag categories, introduced in [§ 2, high-level-methods-for-homotopy-construction-in-associative-n-categories]: Nick Hu2024812https://forest.nickx.hu/coherent-inverses-UGIR/coherent-inverses-UGIR/coherent-inverses-UGIR/zigzag categorydefinition For a category , we define the zigzag category as the category with: objects (zigzags) iterated cospans of ; e.g. for objects , and morphisms of for , The feet of the iterated cospan are called regular levels, while the tips are singular levels. Every zigzag has an associated length, which is its number of singular levels. morphisms (zigzag maps) a collection of morphisms of arranging into a (singular) monotone map of singular levels, combined with a dually determined antiparallel (regular) monotone map of regular levels, such that the induced planar diagram in commutes. E.g. a commuting diagram in of the form: where the morphisms labelled arrange into the singular monotone map , and the morphisms labelled arrange into the (antiparallel) regular monotone map . Monotonicity is essentially the condition that the induced diagram is planar, i.e. that no arrows cross. Composition of zigzag maps is given by taking the composition of along composition of monotone maps (as in ). The identity zigzag map is given by the identity morphisms of equipped with the identity monotone map. is actually a parametric right adjoint functor , which has some interesting consequences that we will explore in due course. Informally, an -fold zigzag category over , , is supposed to be a ‘space’ containing (at least) all the combinatorial encodings of -dimensional string diagrams with respect to some algebraic signature encoded by , whose objects are certain diagrams obtained by stratifying a string diagram into ‘regular’ and ‘singular’ levels (as in the right side of figure ). [§ 3, high-level-methods-for-homotopy-construction-in-associative-n-categories] then goes on to show a method for the construction of complex homotopies by means of a colimit procedure in zigzag categories, as illustrated by figure . This thesis can be summarised as an extension of this construction to framed zigzag enriched categories to equip to the theory a built-in notion of coherently invertible cell, and an enhancement of this procedure to properly support the additional structure. Nick Hu202519https://forest.nickx.hu/coherent-inverses-HTXY/coherent-inverses-HTXY/coherent-inverses-HTXY/Combinatorialised string diagramfigure] (R0R0) to (R0S0); \draw[->] (R0R1) to (R0S0); \draw[->] (R0R1) to (R0S1); \draw[->] (R0R2) to (R0S1); \draw[->] (S0R0) to (S0S0); \draw[->] (S0R1) to (S0S0); \draw[->] (S0R1) to (S0S1); \draw[->] (S0R2) to (S0S1); \draw[->] (R1R0) to (R1S0); \draw[->] (R1R1) to (R1S0); \draw[->] (R1R1) to (R1S1); \draw[->] (R1R2) to (R1S1); \draw[->] (R1R2) to (R1S2); \draw[->] (R1R3) to (R1S2); \draw[->] (S1R0) to (S1S0); \draw[->] (S1R1) to (S1S0); \draw[->] (S1R1) to (S1S1); \draw[->] (S1R2) to (S1S1); \draw[->] (R2R0) to (R2S0); \draw[->] (R2R1) to (R2S0); \draw[->] (R2R1) to (R2S1); \draw[->] (R2R2) to (R2S1); \draw[->] (R0R0) to (S0R0); \draw[->] (R0R1) to (S0R1); \draw[->] (R0R2) to (S0R2); \draw[->] (R0S0) to (S0S0); \draw[->] (R0S1) to (S0S1); \draw[->] (R1R0) to (S0R0); \draw[->] (R1R1) to (S0R1); \draw[->] (R1R3) to (S0R2); \draw[->] (R1S0) to (S0S0); \draw[->] (R1S1) to (S0S1); \draw[->] (R1S2) to (S0S1); \draw[->] (R1R0) to (S1R0); \draw[->] (R1R2) to (S1R1); \draw[->] (R1R3) to (S1R2); \draw[->] (R1S0) to (S1S0); \draw[->] (R1S1) to (S1S0); \draw[->] (R1S2) to (S1S1); \draw[->] (R2R0) to (S1R0); \draw[->] (R2R1) to (S1R1); \draw[->] (R2R2) to (S1R2); \draw[->] (R2S0) to (S1S0); \draw[->] (R2S1) to (S1S1); } \end{tikzpicture} \end{adjustbox} ]]> A 2D string diagram and its combinatorial encoding as an object of .Nick Hu202519https://forest.nickx.hu/coherent-inverses-E4GF/coherent-inverses-E4GF/coherent-inverses-E4GF/Contraction examplefigure Example of a contraction. The nature of this work is particularly exploratory, as there is no existing model with which we can construct a correspondence in the traditional mathematical sense of correctness via a soundness-and-completeness style of argument. However, there are certain nice properties that we would expect to hold, and we can show using the theory we have developed, which provides the basis for an argument of ‘correctness’. This is the subject of chapter . Moreover, as is often the case with formalisation work, the process of formalising higher-dimensional string diagrammatic arguments that are expected to hold (given that they are higher-dimensional variants of known results) additionally provides a means to test the utility of the theory and tooling we develop, which we explore in chapter . Nick Hu2023123https://forest.nickx.hu/coherent-inverses-invertibility/coherent-inverses-invertibility/coherent-inverses-invertibility/Coherent invertibilitysection In a category, we say that objects and are isomorphic when there exists a morphism and an inverse morphism satisfying: In this case, we say that and are an inverse pair, witnessing the isomorphism . This is a proof-relevant notion of ‘sameness’ for and , as there may be other inverse pairs that witness , and is one of the distinguishing aspects between set-theoretic mathematics and category-theoretic mathematics. In other words, a set is a mathematical structure that is fundamentally equipped with an equality predicate on its members; categories build on this, allowing one to express isomorphism between objects, but not ‘isomorphism’ between morphisms. Higher categories address this by allowing for 2-morphisms between (1-)morphisms, 3-morphisms between 2-morphisms, and so on. Naturally, this leads to a generalisation of what it means for a morphism to admit an inverse. In a 2-category, such an inverse arises from replacing the above equalities with directed 2-morphisms that perform pair cancellations or introductions: We may ask that these 2-morphisms themselves form inverse pairs, leading to a coinductive definition of inverse, generating structure in all dimensions. In string diagrams, these 2-morphisms are drawn as ‘cups’ and ‘caps’, representing the introductions and cancellations respectively as in figure . Nick Hu2025114https://forest.nickx.hu/coherent-inverses-B4SI/coherent-inverses-B4SI/coherent-inverses-B4SI/Cups and capsfigure The resulting notion of inverse has poor algebraic properties. To see why, consider the following 2-morphism, obtained as a composite of the 2-morphisms above: Here we underline the redex applying in each case, and assume for simplicity our theory is definitionally unital and associative. This 2-morphism is an instance of a snake composite (see section for another), named as such for its string diagrammatic representation in figure , and has type . We might expect it to be equivalent to ; this must certainly be the case if we desire the free -category generated by an invertible to itself be equivalent to the free -category generated by a single object, where parallel morphisms are always equivalent. However, under the coinductive definition of inverse presented before, this will not be the case; we say that this notion of inverse is incoherent. Nick Hu2025114https://forest.nickx.hu/coherent-inverses-R49Y/coherent-inverses-R49Y/coherent-inverses-R49Y/Snake compositefigure Snake composite In order to have a coherent notion of invertibility, this snake composite must be equivalent to the identity (i.e. the picture in figure should admit a ‘straightening’), via a 3-morphism arising ambiently from the structure of the higher category itself, which we call a higher-dimensional coherence. There is an infinite sequence of such phenomena, in all dimensions, known as catastrophes from their study in manifold theory. section explores some of higher-dimensional coherences in low dimensions in more detail. Any higher-dimensional computer algebra system must have a solution to this problem in order to implement a satisfactory theory of inverses. We approach this in homotopy.io by considering what the inverse of a morphism (or more generally: an -dimensional string diagram) should look like and deducing what its combinatorial representation, as alluded to in section , should be. Diagrammatically, at the level of -morphisms, and admit string diagrammatic representations in a 1D diagrammatic calculus as in figure , where the is represented by the yellow point, is represented by the blue wire, and is represented by the red wire. Composition in this diagrammatic calculus is given by juxtaposition along the vertical axis. Nick Hu2025115https://forest.nickx.hu/coherent-inverses-ILG4/coherent-inverses-ILG4/coherent-inverses-ILG4/Morphisms as 1D string diagramsfigure This suggests that the inverse of should look like a ‘reflection’ of , with respect to that axis. When we consider the combinatorial encodings of , which is given by the natural analogue for is distinguishing the left leg of the cospan from the right (we have rotated the axis of composition with respect to figure to save space). Crucially, we do not use a formal ‘’ token here: we know that infinitely many higher morphisms are required to witness that this inverse is coherent, but the structure of any such coherence is arbitrarily complex and it seems intractible to encode this in this way. A problem arises if instead we had taken some endomorphism , a perfectly valid 1-morphism algebraically, instead of (i.e. forcing ): in this instance, both and its inverse would admit combinatorial representations that coincide as We solve this by equipping to each arrow some data that carries its intrinsic direction, which we call a frame: encoding and respectively, using formal tokens and . We call this extended structure a framed zigzag, and chapter constructs this categorically using techniques from enriched category theory. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-overview/coherent-inverses-overview/coherent-inverses-overview/Overviewsectionchapter presents some general category theory required to understand this thesis, although more specialist topics are introduced within their respective chapters when called upon. Basic category theory, along with some enriched category theory, is assumed throughout. A good reference can be found in [basic-category-theory] for the former, and [basic-concepts-of-enriched-category-theory] for the latter. Familiarity with string diagrams (e.g. for monoidal categories) will also be helpful; see [a-survey-of-graphical-languages-for-monoidal-categories], or [categories-for-quantum-theory-an-introduction] for a textbook account. chapter presents zigzag categories, and their enhancement as framed zigzag enriched categories, which is our main categorical arena of study in this thesis. This begins with a review of the basic notion already present in the literature ([high-level-methods-for-homotopy-construction-in-associative-n-categories], [zigzag-normalisation-for-associative-n-categories], [a-layout-algorithm-for-higher-dimensional-string-diagrams]), and goes on to develop their extension into framed zigzag enriched categories. We do this by capturing the notion as a parametric right adjoint arising from an essentially algebraic theory, which we interpret in -enriched categories, in turn determining its universal properties. chapter shows how to extend the contraction algorithm [§ 3, high-level-methods-for-homotopy-construction-in-associative-n-categories] to framed zigzag enriched categories, and determines its base case as a kind of weighted colimit procedure that uses the additional structure afforded to us by enrichment. We use this to present a correctness argument for our extended algorithm. chapter provides an overview of the implementation of homotopy.io, which implements the theory as a web-based graphical proof assistant. We describe some of the practical engineering considerations that went into the design of the system, and some of the other mechanisms necessary to provide a usable tool for -dimensional string diagrammatic reasoning. chapter presents several case studies that demonstrate the utility of coherently invertible generators within homotopy.io, showcasing newly formalised results that leverage this additional functionality. Additionally, this presents empirical evidence that our construction has good properties. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-related/coherent-inverses-related/coherent-inverses-related/Related worksection The topic of this thesis is part of a larger research programme regarding homotopy.io as a proof assistant for higher-categorical string diagrams, initiated in [associative-n-categories] and [high-level-methods-for-homotopy-construction-in-associative-n-categories], a successor project to the Globular proof assistant. The foundational mechanism underlying Globular is that of manually encoding coherences for a semistrict version of tetracategories, so it is fundamentally limited to -categories for , as opposed to homotopy.io, which works for arbitrary . Moreover, Globular possessed invertibility functionality, which homotopy.io lacked prior to this work; however, this was based on a coinductive but incoherent notion, so lacked the good properties of section . A complement to this thesis is the work of [a-computational-approach-to-higher-categories], which details the underlying theory and implementation of separate aspects of homotopy.io. This includes the rendering subsystem for drawing higher-dimensional string diagrams, and some theory underlying an extended expansion algorithm dual to our contraction algorithm. [homotopy-io-a-proof-assistant-for-finitely-presented-globular-n-categories] is an implementation paper that gives an overview of homotopy.io from a practical perspective. On the side of theory, the notion of zigzag category was introduced in [high-level-methods-for-homotopy-construction-in-associative-n-categories]. [zigzag-normalisation-for-associative-n-categories] outlines a (now defunct) procedure for normalisation of zigzags, which was formerly used for typechecking. [posetal-diagrams-for-logically-structured-semistrict-higher-categories] presents a generalisation of the zigzag construction, orthogonal to ours, to posets in place of the finite linear orders (the augmented simplex category ), with an eye toward providing behaviour to support categorical limit constructions in homotopy.io — although such a theory models something that is necessarily much more general than a string diagram. [manifold-diagrams-for-higher-categories] is a recasting of this theory into more -categorical language in order to formalise the connection between our notions of ‘diagram’ and ‘homotopy’ to their mathematical analogues — in essence, providing a kind of Joyal-Street-esque result establishing the connection between this work and models of -categories. With respect to string-diagrammatic tooling, most tools exist to support various theories of monoidal categories, rather than adopting the higher-categorical perspective that homotopy.io and Globular take. A notable exception is the rewalt toolkit, which is built on the theory of diagrammatic sets. This uses a different approach for semistrictness: rather than Gray semistrictness with strict associators and unitors and weak interchange, which homotopy.io is based on, rewalt is based on so-called ‘Simpson semistrictness’, which has strict interchange and weak unitors instead. As a result of this difference, the resulting theory of string diagrams is quite different in each tool. Another difference is that rewalt is more like a software library than an interactive graphical program, where user interaction is code-based as opposed to clicking-and-dragging on various pictures. A further summary of tools for string diagrams can be found in [§ 1.2, homotopy-io-a-proof-assistant-for-finitely-presented-globular-n-categories]. Yet another approach to computational methods for higher categories is via a type-theoretic presentation, in the form of the type theory CaTT. Semistrict variations of CaTT, which are more closely related to the kind of higher category presented by homotopy.io, have also been studied in [a-type-theory-for-strictly-unital-∞-categories] and [a-syntax-for-strictly-associative-and-unital-∞-categories], and further elaborated in the thesis [a-type-theoretic-approach-to-semistrict-higher-categories]. A definition for what it means for a cell to be invertible in this framework is given by [invertible-cells-in-ω-categories], but higher coherences are not addressed. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-meta/coherent-inverses-meta/coherent-inverses-meta/Metasection This document is available in two formats: an online version, available at https://nickx.hu/thesis, and in PDF form. Throughout, many technical terms are hyperlinked to their definitions. The source code for the accompanying software, homotopy.io, is available at https://github.com/homotopy-io/homotopy-rs. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-background/coherent-inverses-background/coherent-inverses-background/Backgroundchapter Throughout this thesis, we assume a basic understanding of category theory and familiarity with string diagrams as a means to present morphisms of some kind of higher category, especially for chapter and chapter . chapter is best appreciated with some understanding of programming, but no specialist knowledge is required. A similar sentiment holds for chapter , but for low-dimensional topology instead. These chapters can be read fairly independently from the rest of the thesis, for the reader who is more interested in the implementation of a proof assistant or using homotopy.io as a string diagram tool than its underlying theory. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-notation/coherent-inverses-notation/coherent-inverses-notation/notation Every (enriched) category is assumed to be strict and small (often finite), unless otherwise specified, which means that its objects form a set and can be compared for equality. As such, we will deemphasise size issues, as is typical; in our constructions, the usual considerations apply without needing to be explicit, and thus distracting from the technical point to be made. Categories (and enriched categories) are typically denoted by calligraphic script, e.g. , and objects and morphisms are typically denoted by lowercase script, e.g. and respectively, with uppercase script (e.g. ) denoting (enriched) functors. A notable exception is we will often use in ordinary script to denote the (enriched) category which is the index of a diagram, an (enriched) functor, to connote that it is built out of a small (usually finite) amount of generating data. Morphisms are given diagrammatically as , and our string diagrams are to be read bottom-up, left-to-right. In the context of higher categorical concepts, we use the terms ‘-cell’ and ‘-morphism’ interchangeably, although the former with slightly more emphasis on the model of the higher category (e.g. in homotopy.io) as opposed to the higher category itself, and the double-tailed arrow is used for 2-morphisms. and denote initial and terminal objects, and (sometimes with some overlap) denotes the -simplex: the finite linear order ; by confusing convention, this means that and in contrast to the von Neumann ordinals, and the -simplex denotes the empty set. Whenever we index over the natural numbers, we begin at 0. The rest of the categorical preliminaries follow. Nick Hu2024926https://forest.nickx.hu/coherent-inverses-AOTF/coherent-inverses-AOTF/coherent-inverses-AOTF/Sliced adjunctionslemmanLab Let be an adjunction. Then for any object , there is an adjunction where is given by applying , and Dually, in the case that has pullbacks, for any object , there is an adjunction where is given by applying , and given by pullback along the unit of the adjunction. Nick Hu2024611https://forest.nickx.hu/coherent-inverses-JCFF/coherent-inverses-JCFF/coherent-inverses-JCFF/Terminal object of a slice categorylemma In a slice category , the terminal object is equipped with the identity on . Nick Hu2024611proof Every object of is in bijection with a commuting triangle Nick Hu2025523https://forest.nickx.hu/coherent-inverses-ATG5/coherent-inverses-ATG5/coherent-inverses-ATG5/category of elementsdefinition Let be a presheaf. The category of elements is defined by objects for objects of , pairs ; morphisms morphisms of satisfying for , with composition inherited from . This gives rise to a forgetful functor onto the first component of the pair. Nick Hu2024611https://forest.nickx.hu/coherent-inverses-8OTN/coherent-inverses-8OTN/coherent-inverses-8OTN/Slice of a presheaf category is a presheaf categorylemma Let be a presheaf in . There is an equivalence of categories where denotes the category of elements of . Nick Hu2023123https://forest.nickx.hu/coherent-inverses-enriched-category-theory/coherent-inverses-enriched-category-theory/coherent-inverses-enriched-category-theory/Enriched category theorysection Here, we will introduce some basic concepts of enriched category theory. The standard reference is [basic-concepts-of-enriched-category-theory], for the case of enriching over symmetric monoidal closed, but a more modern and condensed treatment (specialised to enriching over cartesian closed, with which we are mostly concerned) can be found in [Appendix A, elements-of-∞-category-theory]. A slight generalisation of this notion, which we occasionally make use of, is enrichment over an arbitrary symmetric monoidal category (which is not required to be closed); an account of this can be found in [Chapter 3, categorical-homotopy-theory]. We will mostly ignore the technical details of this generalisation, as everything works as one might expect, and they are not relevant to the main thrust of this work. Some attention is required for the notion of a change of base for enriched categories, as we frequently make use of the idea that -enriched categories are a ‘shadow’ of 2-categories (-enriched categories). Nick Hu202491https://forest.nickx.hu/coherent-inverses-4OUN/coherent-inverses-4OUN/coherent-inverses-4OUN/change of baselemma[Lemma 3.4.3, categorical-homotopy-theory] Let be a lax monoidal functor. There is an induced 2-functor , called change of base (along ), which takes a -category to a -category . Moreover, is an adjunction of 2-categories if and only if is a monoidal adjunction, if and only if it is strong monoidal, and admits a right adjoint as a functor. Explicitly, the objects of are the same as those of , and the morphisms are given by where the composition, identity, and associator and unitor morphisms are inherited from the monoidal structure of . Nick Hu202491proof The idea that such an assignment extends to a 2-functor is alluded to by [Remark 3.5.11, categorical-homotopy-theory], but a detailed proof can be found in [normed-spaces-and-the-change-of-base-for-enriched-categories]. Among other things, this allows us to create more elaborate enriched categorical structure from functors in ordinary category theory. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-2-categorical-reflections/coherent-inverses-2-categorical-reflections/coherent-inverses-2-categorical-reflections/Reflections of 2-categorical structuresection First, we examine the relationship between categories and partially ordered sets, which themselves can be seen as a special case of categories. Nick Hu202491https://forest.nickx.hu/coherent-inverses-22AL/coherent-inverses-22AL/coherent-inverses-22AL/, the category of partially ordered sets (posets) and monotone functionsdefinition The locally small category has small posets as objects and monotone functions as morphisms. Nick Hu202491https://forest.nickx.hu/coherent-inverses-ZC3N/coherent-inverses-ZC3N/coherent-inverses-ZC3N/ is cartesian closedlemma is a cartesian closed category, where the cartesian product of two posets is their cartesian product as sets, with the product order: The exponential object of two posets is their exponential object as sets, with the pointwise order: Nick Hu202491proof For the closure, we must establish an adjunction That is, a bijection of Hom sets: We can work out the structure of what must be by letting , the terminal object of , to probe its elements, and , the poset , to probe its order. Then deduce that and hence the elements of the poset are monotone functions . Finally, tells us that for , if and only if . We note that is a Bénabou cosmos: in addition to being cartesian closed, it is also complete and cocomplete. This makes a good category to enrich over, and due to closure every poset can be naturally seen as a -category, i.e. the category is enriched over itself. Nick Hu202491https://forest.nickx.hu/coherent-inverses-0X39/coherent-inverses-0X39/coherent-inverses-0X39/ is self-enrichedlemma The set of monotone maps between two posets is itself a poset. Nick Hu202491proof This is true for any cartesian closed category [§ 1.6, basic-concepts-of-enriched-category-theory], and in particular for (lemma ). Explicitly, the set of monotone maps between two posets and is equipped with the pointwise order: whenever for all . We also note that there is a precise sense in which a poset is a special kind of category, characterised by the following universal property. Nick Hu202491https://forest.nickx.hu/coherent-inverses-ZEVP/coherent-inverses-ZEVP/coherent-inverses-ZEVP/ is a reflective subcategory of proposition The inclusion functor , which takes every poset to a thin category, admits a left adjoint Nick Hu202491proof Define the functor by its action on small categories , with being the poset of bidirected cliques of objects in : maximal subsets of objects with the property that there is a morphism between every object, in both directions. In other words, the elements of are the objects of , and the order relation is given by the existence of a morphism between them. This determines a bijection of Hom sets because a functor is determined by its object map, which maps objects of to elements of (the morphism map is uniquely determined by thinness); this is precisely the data of a monotone map between bidirected cliques of objects of and elements of . It is precisely the fact that in proposition is a strong monoidal (in this case, finite-product-preserving) functor that allows us to see a -category as some kind of posetal reflection of a 2-category. This also formally presents the sense in which a poset is a ‘shadow’ of a category. We can also think about the concept of ‘localisation’ in a poset, which is more explicit than its categorical analogue (because morphisms are property not structure). Nick Hu202495https://forest.nickx.hu/coherent-inverses-72BQ/coherent-inverses-72BQ/coherent-inverses-72BQ/Localisation of a posetdefinition Given some poset and elements , the poset localisation of at is the poset induced by adjoining to . Explicitly, is a quotient of where every such that is identified. This notion naturally extends to a class of orderings for which we write . Moreover, we have two ways to turn a -category into a category. Nick Hu2024112https://forest.nickx.hu/coherent-inverses-ZGDQ/coherent-inverses-ZGDQ/coherent-inverses-ZGDQ/The free -category functor is left and right adjointlemma There is an adjunction where sends a category to a -category with only trivial 2-cell filler (i.e. ), sends a -category to a category, quotienting morphisms by connected components in Hom posets, and does so similarly by forgetting all 2-cell fillers. Nick Hu2024112proof This arises as due to the adjoint triple where equips a set with the trivial order, sends a poset to its set of connected components, and forgets ordering. Because both left-adjoint functors and are strong monoidal functors (both preserve cartesian products), they are automatically monoidally adjoint, and hence induce an adjunction in between categories of -enriched categories. This is analogous to the adjoint triple between and induced by the forgetful functor, against equipping a set with the discrete or indiscrete topology (recall that, for finite sets, behaves like the category of spaces: is equivalent to the full subcategory of consisting of Alexandroff spaces via the specialisation topology, and every finite topological space is Alexandroff). Nick Hu2023123https://forest.nickx.hu/coherent-inverses-poset-completion/coherent-inverses-poset-completion/coherent-inverses-poset-completion/The finite non-empty meet completion of a posetsection An important property that a poset may possess is that of ‘having (finite, non-empty) meets’, i.e. being a -semilattice. Nick Hu202491https://forest.nickx.hu/coherent-inverses-BR3I/coherent-inverses-BR3I/coherent-inverses-BR3I/Meet definition Given a poset , the meet of two elements , when it exists, is the element of which is the greatest lower bound of and , denoted by . It is characterised by the following properties: and ; if and , then . When is closed under as a binary operation, is associative, commutative, and idempotent. In this case, we describe as a -semilattice. Note that in a -semilattice , the partial order relation is completely determined algebraically by — that is, a -semilattice is equivalently described by a set equipped with a commutative idempotent semigroup structure. For any non-empty finite subset , the meet of is the greatest lower bound of all elements in , denoted by . The dual operation, the join , is defined similarly. Nick Hu202491https://forest.nickx.hu/coherent-inverses-4UFD/coherent-inverses-4UFD/coherent-inverses-4UFD/Poset with some but not all meetsexample Consider the poset with elements Each pair in has a meet, but there is no meet of all three elements; adding this would make this poset a -semilattice. These -semilattices arrange themselves into a cartesian (and thus symmetric monoidal) category. Nick Hu202491https://forest.nickx.hu/coherent-inverses-QIKX/coherent-inverses-QIKX/coherent-inverses-QIKX/, the category of semilattices with finite non-empty meetsdefinition is the category of -semilattices where morphisms preserve meets: for Note that is automatically monotone with respect to the partial order on and . Similarly, there is a category of -semilattices with finite non-empty joins. Nick Hu202491https://forest.nickx.hu/coherent-inverses-JM4B/coherent-inverses-JM4B/coherent-inverses-JM4B/ is cartesianlemma is cartesian, with the product -semilattice of and given by their cartesian product on underlying sets, equipped with the product meet: Nick Hu20241212https://forest.nickx.hu/coherent-inverses-L0QU/coherent-inverses-L0QU/coherent-inverses-L0QU/Free -completion of a posetlemma The functor which sends a poset to its set of non-empty downwards-closed subsets, ordered by inclusion, with joins given by union, is left adjoint to the forgetful functor . Nick Hu20241212https://forest.nickx.hu/coherent-inverses-F8LS/coherent-inverses-F8LS/coherent-inverses-F8LS/ is strong monoidallemma The functor is strong monoidal (i.e. preserves finite products). Nick Hu20241212proof Here, it is crucial that the action of takes non-empty downwards-closed subsets, or else the terminal poset would not be preserved. The necessary isomorphisms are easy to check. Nick Hu20241212https://forest.nickx.hu/coherent-inverses-QRFA/coherent-inverses-QRFA/coherent-inverses-QRFA/lemma There is a categorical equivalence given by reversing ordering. Nick Hu20241212https://forest.nickx.hu/coherent-inverses-7AXW/coherent-inverses-7AXW/coherent-inverses-7AXW/Adjunction corollary There is an adjunction Explicitly, sends a -category to an -category with the same objects, but every Hom poset is replaced by a -semilattice given by non-empty downwards-closed subsets, reverse-ordered by inclusion. sends an -category to a -category with the same objects, but every Hom lattice forgets its meet and reverses its order. Nick Hu20241212proof Use lemma with lemma , lemma , and lemma . corollary is useful in particular because it provides a formal way to treat a -category as a -category, which we make use of extensively in chapter . Nick Hu2023123https://forest.nickx.hu/coherent-inverses-graphs/coherent-inverses-graphs/coherent-inverses-graphs/Category theory via graphssection This thesis, being both about category theory and its representation in a computer proof assistant, is naturally concerned with the representation of (enriched) categories and functors. Graphs are a natural way to represent category theoretic concepts (e.g. whenever one comes across a commutative diagram), and moreover they are a fundamental data structure in computer science, so they are a natural tool to use for this purpose. Thus, in this section, we introduce a graph-theoretic language to capture (enriched) functors as data, using a notion of ‘weighted graph’ that is to be made precise. For us, all graphs are directed, and not necessarily simple (meaning that the edge relation is boolean), nor loop-free (meaning the edge relation is irreflexive), unless otherwise stated (i.e. these correspond to ‘multigraphs’ in the literature). Nick Hu2023123https://forest.nickx.hu/coherent-inverses-graphs-presheaf-topoi/coherent-inverses-graphs-presheaf-topoi/coherent-inverses-graphs-presheaf-topoi/Graphs as presheaf topoisection There is a correspondence between graphs and categories, which we first sketch here. Nick Hu2024610https://forest.nickx.hu/coherent-inverses-D66F/coherent-inverses-D66F/coherent-inverses-D66F/Topos of reflexive graphsdefinition Let denote the category generated by where is a common retract of and , i.e. . The topos of reflexive graphs is the category of presheaves (functors ): Reflexivity means that for every vertex , there is a distinguished loop although there will be possibly more self-loops at . Typically, we will elide these distinguished loops from diagrams, but note that they allow for a reflexive graph homomorphism to notionally map edges to vertices. We will also normally number vertices to distinguish them, and represent edge names as a pair of numbers when disambiguation is required. Nick Hu2024610https://forest.nickx.hu/coherent-inverses-7H54/coherent-inverses-7H54/coherent-inverses-7H54/example Consider the reflexive graph It has two loops on ; the reflexive loop, and the loop shown. There are two distinct edges from to , and no edges between and any other vertex. It corresponds to a functor in the following way: such that , etc. As a presheaf topos, enjoys many nice properties, including the existence of all small limits and colimits (which are computed pointwise). In general, for a reflexive graph presented in this way, is its set of vertices and its set of edges. Nick Hu2024612https://forest.nickx.hu/coherent-inverses-NL0N/coherent-inverses-NL0N/coherent-inverses-NL0N/Representable presheaves of remark The two representables of are: The trivial reflexive graph with a single node because and . The smallest reflexive graph consisting of a single edge (and reflexive loops for its endpoints) because and . That every presheaf is a colimit of representable presheaves says that every reflexive graph is obtained by gluing together vertices and edges. Homomorphisms between these reflexive graphs are defined by a pair of functions that map vertices to vertices, and edges to edges, in a way that respects the source, target, and reflexivity maps. This is encoded by what it means to be a natural transformation between functors . Nick Hu2024610https://forest.nickx.hu/coherent-inverses-JTJ4/coherent-inverses-JTJ4/coherent-inverses-JTJ4/example Consider the reflexive graphs There are two distinct reflexive graph homomorphisms , respectively sending the edge to the upper and lower edges between and (both of which induce a vertex mapping ), and yet another two distinct reflexive graph homomorphisms which send the edge to the reflexive edges (elided) on and respectively (thus forcing the vertex mapping to be the constant map on or as appropriate). Any surjective mapping of edges fails to be part of a reflexive graph homomorphism, as and hence there is no consistent mapping for the vertex . is intrinsically related to , the category of small categories, by a monadic adjunction. Nick Hu2024611https://forest.nickx.hu/coherent-inverses-G1TF/coherent-inverses-G1TF/coherent-inverses-G1TF/Adjunction between and proposition There is a monadic free-forgetful adjunction For a category , its underlying reflexive graph is denoted by with vertices and edges given by objects and morphisms of (reflexive edges are determined by identity morphisms). Conversely, any reflexive graph admits a free category with objects corresponding to vertices of , and morphisms given by paths in . That this adjunction is monadic (i.e. that is monadic over ) says the following: the data of a category is given precisely by an algebra over the monad induced by this adjunction; in other words, every category is presented by its underlying reflexive graph (generators) equipped with a collection of identifications between parallel paths (relations, which determine equality in each Hom set). This is analogous to being monadic over , which says that groups can be presented by generators and relations. Nick Hu2024611https://forest.nickx.hu/coherent-inverses-T5F8/coherent-inverses-T5F8/coherent-inverses-T5F8/Nerves, truncated simplicial/globular setsremark Readers familiar with combinatorial models of higher category theory may recognise as the 1-truncated simplex category , or equivalently the 1-truncated reflexive globe category . This makes equivalent to 1-truncated simplicial/reflexive globular sets. Additionally, the adjunction observed in proposition can be seen as a restriction of the nerve/realisation adjunction between and . Nick Hu20241113https://forest.nickx.hu/coherent-inverses-DDD6/coherent-inverses-DDD6/coherent-inverses-DDD6/Variants of remark If instead of , we considered presheaves over , we would obtain the category of (not necessarily reflexive) graphs. These can be seen as 1-truncated semi-simplicial sets. Irreflexive (loop-free) graphs can be obtained from this category by taking the full subcategory containing only the presheaves such that Moreover, we can isolate a category of simple (having at most one edge between any pair of vertices) graphs of these categories by taking the full subcategory containing only the presheaves such that is a monomorphism in . We will use the notation to talk agnostically about any of these categories, making it clear from context to which one we are referring. Moreover, without additional qualification, a simple graph is understood to be irreflexive. We are interested in graphs that present functors, carrying ‘typed’ information. This can be modelled by equipping to a graph a graph weighting, assigning to each edge and vertex a label. Nick Hu2024610https://forest.nickx.hu/coherent-inverses-OBWC/coherent-inverses-OBWC/coherent-inverses-OBWC/graph weightingdefinition A graph weighting is a set of vertex weights , and a set of edge weights , such that edge weights have both a source and target vertex weight associated to them, and in the reflexive case vertex weights have an associated self-loop edge weight. We write to denote that the edge label has source vertex label and target vertex label , and to denote the distinguished self-loop edge label associated to the vertex label . Every graph weighting admits a canonical graph, where is given by and is given by , which we also denote by . In the case of simple graphs, the canonical graph is not necessarily simple: instead, we let be given by the set of subsets of which ‘match-up’ on source and targets: Nick Hu2024610https://forest.nickx.hu/coherent-inverses-JGG7/coherent-inverses-JGG7/coherent-inverses-JGG7/weighted graphsdefinition A graph weighting in is a graph homomorphism . Moreover, the slice category is the category of graphs equipped with a graph weighting in (-graphs), with morphisms being homomorphisms of -graphs which respect the graph weighting , of which is the terminal object. In the case of simple graphs, we instead consider the full subcategory of the slice over the non-simple category of graphs whose projection onto domain is a simple graph. Nick Hu2024624https://forest.nickx.hu/coherent-inverses-IF61/coherent-inverses-IF61/coherent-inverses-IF61/Reflexively weighted edgesremark Note that in a -reflexive-graph, every vertex has a self-loop which has the reflexive weight on its vertex weight; however, there may be other edges which also have this weight. For example, the -reflexive-graphs are distinct (the first has one vertex and the second has two). As before, we elide the distinguished reflexive self-loop on each vertex. We use subscripts to denote the vertex indices of -graphs. Nick Hu2024611https://forest.nickx.hu/coherent-inverses-X0XS/coherent-inverses-X0XS/coherent-inverses-X0XS/Explicit -reflexive-graphexample Let , and constitute a graph weighting. As a reflexive graph, looks like: An example -reflexive-graph is the following: where the implied homomorphism is made evident by our notation. Now we apply the theory we have developed, in order to provide a concrete graph-theoretic presentation of -functors. We focus on -functors for ease of presentation, but the theory works equally well for ordinary functors in the obvious way. Nick Hu2024106https://forest.nickx.hu/coherent-inverses-RISV/coherent-inverses-RISV/coherent-inverses-RISV/-graph of a -functordefinition Let be a -functor between small -categories, i.e. a morphism of . is an object of the slice category , which by lemma , lemma , and proposition is part of an adjunction is the category of -reflexive-graphs, where is the graph weighting given by vertices and edges of the underlying graph of the underlying category of , i.e. the objects and morphisms of , forgetting 2-cell fillers. The image of is a particular -reflexive-graph in , which we call the -reflexive-graph of . By not explicitly representing distinguished self-loops, we can instead work in a slice category of , and additionally if we know that has no non-trivial endomorphisms then we can simplify further to . By additionally identifying two -functors , where is a wide sub--category of , whenever i.e. may have more morphisms than , but not in a way that is detectable by applying (equivalently: is a retract of in ), we can represent this class as a -simple-graph, where the vertices are given by objects of , weighted by objects of , and the edge weights are given by with edges existing whenever such a set is non-empty. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-framed-zigzags/coherent-inverses-framed-zigzags/coherent-inverses-framed-zigzags/Framed zigzag enriched categorieschapter In this chapter, we first review the zigzag construction as described by [high-level-methods-for-homotopy-construction-in-associative-n-categories], and [zigzag-normalisation-for-associative-n-categories]. Subsequently, we recast it in the abstract setting of parametric right adjoint functors, making it amenable for enrichment so that it can support a theory of coherently invertible generators. The description of the zigzag functor as a polynomial functor is due to [manifold-diagrams-for-higher-categories], but otherwise all the material after the first section is new — to appear in [coherent-invertibility-in-associative-n-categories]. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-zigzags-explicitly/coherent-inverses-zigzags-explicitly/coherent-inverses-zigzags-explicitly/The zigzag category, explicitlysection In this section, we recall the basic definitions required to construct the zigzag category explicitly. Many of these definitions can be found in [high-level-methods-for-homotopy-construction-in-associative-n-categories], but we follow [zigzag-normalisation-for-associative-n-categories] in that we slightly generalise to the case where the annotations on regular monotone maps are not required to be identity morphisms (globularity). The combinatorics of these gadgets are heavily related to the augmented simplex category, which is commonly seen in shape-based models of higher categories. First, we recall Wraith's duality between finite linear orders and the opposite category of finite intervals. Nick Hu2024812https://forest.nickx.hu/coherent-inverses-2U9F/coherent-inverses-2U9F/coherent-inverses-2U9F/augmented simplex categorydefinition The augmented simplex category is the category with: objects finite ordinals, denoted by for , or sometimes pictorially: e.g. as or etc. morphisms monotone maps. Such a category is ‘augmented’ as it includes an initial object (the empty ordinal). Nick Hu2024812https://forest.nickx.hu/coherent-inverses-5OOR/coherent-inverses-5OOR/coherent-inverses-5OOR/finite interval categorydefinition The finite interval category is the category with: objects finite ordinals for , or equivalently, finite ordinals which admit top and bottom elements. morphisms top-and-bottom-preserving monotone maps. Nick Hu2024812https://forest.nickx.hu/coherent-inverses-WFIU/coherent-inverses-WFIU/coherent-inverses-WFIU/Singular/regular monotone dualitylemma[using-the-generic-interval] There is an equivalence of categories between the category of augmented simplices and the opposite category of finite intervals: Nick Hu2024812proof We will establish that the functor is an equivalence of categories, where is a reinterpretation of the contravariant Hom functor on . First, can be given that type because, for any , the set of monotone maps can be identified with . This is because any such monotone map into is fully determined by the (unique by monotonicity) such that . Pictorially, drawing this is choosing as one of the , of which there are possibilities. This also determines that is (essentially) surjective. It remains to show that it is also fully faithful. Its action on a monotone map is precomposition: that is, given some encoding of a monotone map , we can obtain some encoding of a monotone map . As we have made an identification between encodings of monotone maps and simplices , we can describe this action graphically — e.g. for and given below: that is, the mapping on Hom sets induced by , is bijective, as the monotone map upwards between s determines (and is equivalently determined by) the monotone map downwards between s. Nick Hu2024812https://forest.nickx.hu/coherent-inverses-MQNI/coherent-inverses-MQNI/coherent-inverses-MQNI/Wraith's duality as a simplified Stone dualityremark As described by [using-the-generic-interval], lemma can be seen as the restriction of Stone duality to the case of finite linear orders. Next, we recall the most explicit definition of the ordinary (unenriched) zigzag category. Nick Hu2024812https://forest.nickx.hu/coherent-inverses-UGIR/coherent-inverses-UGIR/coherent-inverses-UGIR/zigzag categorydefinition For a category , we define the zigzag category as the category with: objects (zigzags) iterated cospans of ; e.g. for objects , and morphisms of for , The feet of the iterated cospan are called regular levels, while the tips are singular levels. Every zigzag has an associated length, which is its number of singular levels. morphisms (zigzag maps) a collection of morphisms of arranging into a (singular) monotone map of singular levels, combined with a dually determined antiparallel (regular) monotone map of regular levels, such that the induced planar diagram in commutes. E.g. a commuting diagram in of the form: where the morphisms labelled arrange into the singular monotone map , and the morphisms labelled arrange into the (antiparallel) regular monotone map . Monotonicity is essentially the condition that the induced diagram is planar, i.e. that no arrows cross. Composition of zigzag maps is given by taking the composition of along composition of monotone maps (as in ). The identity zigzag map is given by the identity morphisms of equipped with the identity monotone map. It will transpire that the action of taking a category to its category of zigzags is functorial, but we will prove this more abstractly later rather than with the explicit definition of definition . Nick Hu2024812https://forest.nickx.hu/coherent-inverses-BWBW/coherent-inverses-BWBW/coherent-inverses-BWBW/Unlabelled zigzag categoryexample For the terminal category , the category is the category with objects: where every morphism is the identity. Zigzag maps are given by a monotone map between singular levels, trivially equipped with a collection of morphisms of . In other words, there is an equivalence of categories Nick Hu2024812https://forest.nickx.hu/coherent-inverses-9V13/coherent-inverses-9V13/coherent-inverses-9V13/singular projectiondefinition Every zigzag category is naturally equipped with a singular projection functor , which sends each zigzag to where is the number of singular levels of (its length). Each zigzag map is sent to its underlying monotone map between singular levels. Again, it will later become apparent that the singular projection functor is the image of the terminal functor in the functor , along the equivalence observed in example . Under certain conditions, the singular projection admits certain opcartesian lifts: informally, this allows us to compute colimits in a zigzag category by computing them in and lifting each fibre [§ 3, high-level-methods-for-homotopy-construction-in-associative-n-categories], although we shall derive an alternative means to realise this fact in section . Nick Hu2023123https://forest.nickx.hu/coherent-inverses-zigzags-abstractly/coherent-inverses-zigzags-abstractly/coherent-inverses-zigzags-abstractly/The zigzag category, abstractlysection The zigzag category described explicitly is nice for the purposes of implementing a proof assistant, but it leaves much to be desired from a theoretical perspective. For instance, we did not show that is a functor, nor did we establish what the universal property of is, if any. In order to do this, we first recast the ordinary zigzag category construction in a more abstract way, which simultaneously addresses these concerns and moreover provides a foundation that will generalise more readily to the enriched setting. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-zigzags-polynomial/coherent-inverses-zigzags-polynomial/coherent-inverses-zigzags-polynomial/Via polynomial functorssection A first observation is that all the combinatorics of the zigzag category can be packaged into a single gadget, which we can do equivalently in two ways: firstly, as a polynomial functor, and then more generally as a parametric right adjoint. Nick Hu202489https://forest.nickx.hu/coherent-inverses-2H06/coherent-inverses-2H06/coherent-inverses-2H06/parametric right adjointdefinition Let be a functor, such that admits a terminal object . factors as a composite where the second functor is the forgetful projection onto the domain. When is a right-adjoint functor: then we say that is a parametric right adjoint. By using the observation that , one can realise the zigzag functor as an instance of a parametric right adjoint. Nick Hu2024811https://forest.nickx.hu/coherent-inverses-SKZ7/coherent-inverses-SKZ7/coherent-inverses-SKZ7/zigzag functor as a parametric right adjointdefinition The zigzag functor is the parametric right adjoint given by where is essentially given by the zigzag category, equipped with the singular projection. We have a proof obligation: we must show that indeed is a right adjoint. One way to do this is to observe that, in definition , a zigzag map has two constituents: its underlying monotone map between singular levels (shape) and a collection of morphisms in the category over which zigzags are taken (type). We can formally separate the two by defining a ‘universal bundle’ for zigzags (providing the shape) and ‘taking pullbacks’ to combine this with information expressing type. By analogy, taking the category as a kind of ‘classifying space’, the ‘universal -bundle’ is given by the forgetful functor from pointed sets to sets, through which we find a bijective correspondence between discrete fibrations and presheaves by pullback along : Any discrete fibration can be identified (up to isomorphism) as a pullback along of some presheaf, so in this sense the universal -bundle classifies the discrete fibrations. The categorification of this, replacing with , is called the Grothendieck construction, which classifies the (not necessarily discrete) fibrations through a bicategorical equivalence with -valued presheaves. For us, the idea is that will act as the classifying space below the universal zigzag bundle, allowing us to classify the zigzag categories, albeit with more machinery than a mere pullback. Nick Hu202476https://forest.nickx.hu/coherent-inverses-V0EG/coherent-inverses-V0EG/coherent-inverses-V0EG/universal zigzag bundledefinition Let be the category with: objects morphisms sets of monotone maps where is the map which sends a monotone map to its dual monotone map , as in lemma . Composition is given by composition of monotone maps. There is a canonical functor that sends and to , the universal zigzag bundle, which is exponentiable in . Nick Hu2024812https://forest.nickx.hu/coherent-inverses-6EJS/coherent-inverses-6EJS/coherent-inverses-6EJS/ is a right adjointproposition (definition ) is a right adjoint. Nick Hu2024812proof The universal zigzag bundle is exponentiable, so its induced pullback functor admits a right adjoint: Moreover, every pullback functor admits a left adjoint. In particular, so does the pullback functor induced by the terminal functor : Furthermore, as is terminal, there is an (adjoint) equivalence of categories Composing these adjunctions we can deduce is a composite of right adjoints and therefore a right adjoint. Furthermore, this allows one to express the entire zigzag functor as a polynomial functor. Nick Hu202476https://forest.nickx.hu/coherent-inverses-7TQX/coherent-inverses-7TQX/coherent-inverses-7TQX/zigzag polynomial functordefinition is the polynomial functor determined by over the polynomial This construction allows us to package up all the combinatorics associated to the zigzag construction into one category. Moreover, because adjoints universally determine each other, it also would have been possible to give the construction by defining . Next, we will show how to obtain this as a left Kan extension of some small generating presentation, by the observation that and are categories of models of essentially algebraic theories — in particular, they are both cocomplete, and we can find small dense subcategories of them. We opt to obtain this construction abstractly, determining it up to its universal properties, although we do recover an explicit description in example which we shall use in subsequent chapters. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-density/coherent-inverses-density/coherent-inverses-density/The theory of densitysection We first recap the theory of density, as presented in [Chapter 5, basic-concepts-of-enriched-category-theory]. Nick Hu2024815https://forest.nickx.hu/coherent-inverses-RQH3/coherent-inverses-RQH3/coherent-inverses-RQH3/dense functordefinition A functor is dense if its relative nerve is fully faithful. Density is a property that comes in many forms; in essence, it says that every object of is canonically a colimit of a diagram of objects of (assuming such a functor is a subcategory inclusion) — that is some small generating presentation for . Moreover, the relative nerve, under mild assumptions, is universally characterised as the right adjoint to left Kan extension along the Yoneda embedding. Nick Hu2024817https://forest.nickx.hu/coherent-inverses-Q6LL/coherent-inverses-Q6LL/coherent-inverses-Q6LL/relative nerve is right adjointlemma[Proposition 3.2.2, coend-calculus] Given any functor , for which is small and is locally small and cocomplete, its relative nerve admits a left adjoint via a left Kan extension: where is the Yoneda embedding of . For a fully faithful dense functor, the relative nerve is a decomposition of the Yoneda embedding in a certain sense. Nick Hu2024821https://forest.nickx.hu/coherent-inverses-SLV1/coherent-inverses-SLV1/coherent-inverses-SLV1/Yoneda embedding via relative nervelemma When is fully faithful, the composite is naturally isomorphic to the Yoneda embedding . Nick Hu2024821proof For any , calculate We are interested in particular dense functors of and some variants, which we will now explore. Nick Hu2024815https://forest.nickx.hu/coherent-inverses-71HA/coherent-inverses-71HA/coherent-inverses-71HA/dense subcategory of proposition Let be the subcategory of : where the three arrows are the three non-degenerate inclusions. is dense. Nick Hu2024815proof This essentially holds because the nerve of every category is a 2-coskeletal simplicial set, and vice versa, but we will prove this directly also. The nerve relative to , , sends each category to the presheaf . For this functor to be fully faithful, its morphism mapping with respect to a pair of categories and needs to be a bijection: That is, every functor must be in bijective correspondence with a natural transformation . Each such natural transformation is a family of maps for and naturality conditions. Note that is equivalent to the set of objects of , and similarly for ; similarly, an element of is a choice of a morphism of , and that of a composable pair of morphisms (along with its composite). So each component of is respectively a map of objects of to objects of , a map of morphisms of to morphisms of , a map of composable pairs of morphisms of to composable pairs of morphisms of . Therefore, the presence of and witness faithfulness: the data of a functor is precisely its object map and its morphism map. Fullness, the condition that every such natural transformation determines a functor, is given by the naturality conditions imposed by the remaining data of : aside from just mapping objects to objects and morphisms to morphisms, a functor must also be consistent with respect to the source and target of morphisms: i.e. the morphism mapped to must have domain (witnessed by the inclusion ), and codomain (witnessed by the inclusion ); preserve identity, witnessed by the retract ; preserve composition, witnessed by the three non-degenerate inclusions . In some sense, this is an alternative formalisation of the monadic adjunction of proposition — that a category is an algebraic gadget whose generators are reflexive graphs. We are typically interested in cases where our dense functor is also fully faithful; indeed, the straightforward extension of proposition to the full subcategory spanned by is one such example. Nick Hu2024823https://forest.nickx.hu/coherent-inverses-4WLB/coherent-inverses-4WLB/coherent-inverses-4WLB/dense subcategory of a slice categoryproposition Suppose that we have a dense subcategory . For any object , the projection functor is dense. Moreover, when is fully faithful, so is . Nick Hu2024823proof Consider the diagram where the isomorphism in the top right corner is given by lemma . The left square is a pullback: objects of the pullback are a pair of and such that the domain of the morphism is , but this precisely is the category . The outer square is a pullback: objects are a pair of and a natural transformation . Moreover, is forced to be a presheaf of the form , so the type of is . Because is dense, its relative nerve is fully faithful, and hence we deduce an isomorphism for any pair of objects and in : In particular, is a morphism in , and indeed this fibre product is isomorphic to . By the pullback lemma, the right square is also a pullback. Fully faithful functors are preserved by pullback, so fully faithfulness of implies that of , and hence we deduce is dense as required. The left square being a pullback also allows us to deduce that is fully faithful if is. This tells us how to find a dense subcategory of . Nick Hu2024829https://forest.nickx.hu/coherent-inverses-01JA/coherent-inverses-01JA/coherent-inverses-01JA/dense subcategory of proposition Let be the dense subcategory of as in proposition . Then, by proposition , the category of elements is a dense subcategory of . Explicitly, this category has as objects pairs where and is a functor. Alternatively, an object of is one of the following: an object of , i.e. a particular ordinal corresponding to ; a monotone map corresponding to ; a pair of composable monotone maps corresponding to . Let denote the dense full subcategory of which this induces. Dense functors are a kind of generalisation of an essentially algebraic theory, giving a certain perspective on algebraic structure (such as the essence of the zigzag category that we aim to capture). Nick Hu2023123https://forest.nickx.hu/coherent-inverses-essentially-algebraic-theories/coherent-inverses-essentially-algebraic-theories/coherent-inverses-essentially-algebraic-theories/essentially algebraic theoriessectionDense functors allow for a categorical perspective on essentially algebraic theories, which subsume many categorical constructions which are algebraic in nature, including categories themselves and our zigzag construction. Intuitively, the idea that categories are an essentially algebraic theory is a formalisation of the idea that a category can be thought of as a class of objects and morphisms, equipped with some partially defined composition operation ​ satisfying some properties (composition is unital and associative). Many definitions and results here are from [Chapter 5, basic-concepts-of-enriched-category-theory]. Nick Hu2024815https://forest.nickx.hu/coherent-inverses-XIP0/coherent-inverses-XIP0/coherent-inverses-XIP0/essentially algebraic theorydefinition[§ 5.13, basic-concepts-of-enriched-category-theory] An essentially algebraic theory is a dense functor where is small and is cocomplete. For the remainder of this section, will denote an essentially algebraic theory unless stated otherwise. Nick Hu2024816https://forest.nickx.hu/coherent-inverses-XO51/coherent-inverses-XO51/coherent-inverses-XO51/model of an essentially algebraic theorydefinition A model of an essentially algebraic theory is given by a functor such that its opposite functor admits a relative nerve which factorises (up to isomorphism) These models form a full subcategory of the presheaf category : We will not concern ourselves with this lifting condition, as it arises by virtue of being cocomplete. Nick Hu2024817https://forest.nickx.hu/coherent-inverses-9LZ1/coherent-inverses-9LZ1/coherent-inverses-9LZ1/Characterisation of models of an essentially algebraic theorylemma[Theorem 5.59, basic-concepts-of-enriched-category-theory] Because is cocomplete, we have an equivalence of categories That is, every model of an essentially algebraic theory is just an object of . In fact, we can generalise this away from (this time, working with the dual of a model — a comodel) to any arbitrary cocomplete category ​ . Nick Hu2024816https://forest.nickx.hu/coherent-inverses-PCO8/coherent-inverses-PCO8/coherent-inverses-PCO8/Internal -coalgebradefinition An internal -coalgebra in a cocomplete category is a functor which admits a lifting of relative nerves (up to isomorphism) This equivalently expresses that admits a left -relative right adjoint : , defined by a natural bijection Synonymously, we refer to (the image of) this functor as a co- internal to . The category of internal -coalgebras in is denoted by For example, if , then a co-category internal to is given by a functor satisfying the above. We will seek to capture the condition of being an internal coalgebra as the preservation of certain classes of colimits, using the technique of density presentations. Nick Hu2024823https://forest.nickx.hu/coherent-inverses-QG94/coherent-inverses-QG94/coherent-inverses-QG94/density presentationdefinition A density presentation for a fully faithful dense functor is a family of diagrams such that for each , the colimit of exists and is preserved by for each , and moreover is the closure of under this family of colimits. This notion is a simplified version of that of [§ 5.4, basic-concepts-of-enriched-category-theory], restricted to the case where . Nick Hu2024823https://forest.nickx.hu/coherent-inverses-7UUA/coherent-inverses-7UUA/coherent-inverses-7UUA/Canonical density presentationproposition[Theorem 5.29, basic-concepts-of-enriched-category-theory] Every fully faithful dense functor admits a canonical density presentation, given by Nick Hu2024823https://forest.nickx.hu/coherent-inverses-ZB0A/coherent-inverses-ZB0A/coherent-inverses-ZB0A/density presentation of lemma The dense full subcategory inclusion of admits a density presentation given by the diagram . Nick Hu2024823proof This arises as the Segal condition for ordinary categories: a 2-coskeletal simplicial set is the nerve of some category precisely when it satisfies this condition, i.e. each of its 2-simplices have unique horn fillers. Nick Hu2024825https://forest.nickx.hu/coherent-inverses-UJQI/coherent-inverses-UJQI/coherent-inverses-UJQI/-coalgebracorollary An internal -coalgebra in a category is precisely a functor which preserves the colimit Nick Hu2024829https://forest.nickx.hu/coherent-inverses-I6QY/coherent-inverses-I6QY/coherent-inverses-I6QY/density presentation of corollary The dense full subcategory inclusion of admits a density presentation given by diagrams of the form Nick Hu2024829https://forest.nickx.hu/coherent-inverses-TUXL/coherent-inverses-TUXL/coherent-inverses-TUXL/-coalgebracorollary An internal -coalgebra in a category is precisely a functor which preserves the colimits The crux of this section is to give a method of expressing a left adjoint functor as an internal coalgebra, as follows. Nick Hu2024817https://forest.nickx.hu/coherent-inverses-87R5/coherent-inverses-87R5/coherent-inverses-87R5/internal coalgebras correspond to left adjointsproposition[Theorem 5.56, basic-concepts-of-enriched-category-theory] Let denote the subcategory of the functor category consisting of left adjoints. When is cocomplete, there is an equivalence of categories where the functor is restriction along . Nick Hu2024817proof This result comes from [Theorem 5.56, basic-concepts-of-enriched-category-theory], but we can also give a direct calculation using coend calculus as follows. Assuming that is an internal coalgebra; we will show that is a left adjoint to some . Calculate Now suppose that we have an adjunction ; the restriction of along is an internal coalgebra because directly from the adjunction, and hence as required. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-internal-coalgebra/coherent-inverses-internal-coalgebra/coherent-inverses-internal-coalgebra/zigzag category as an internal coalgebrasectionproposition gives a very fast way to specify left adjoint functors, in terms of an internal coalgebra in the codomain category, with respect to an essentially algebraic theory, which presents the theory of the domain, as we will see in example . Nick Hu2024817https://forest.nickx.hu/coherent-inverses-462J/coherent-inverses-462J/coherent-inverses-462J/Left adjoint functors as co-graphsexample The dense subcategory of proposition restricts to a dense subcategory , by taking the full subcategory not containing (it is no coincidence that this is precisely the category of definition , as a category is in some sense a reflexive graph equipped with a composition operation). In this case, is merely the (fully faithful) Yoneda embedding. Because is cocomplete, by proposition , every left adjoint functor is equivalent to an internal co-graph in . Such data is given by any functor , which automatically satisfies the lifting property of definition ; this is because the relative nerve along the Yoneda embedding is isomorphic to the identity functor on . Let us realise the free category functor as in this manner. Define sending to the terminal category , and to the walking arrow category , with the obvious map on morphisms. Now, is computed as the left Kan extension . As is cocomplete, this Kan extension is determined pointwise with the following colimit formula: where the comma category is the category of elements of the functor , and is its forgetful first projection. has objects pairs ; that is, for it is simply a graph homomorphism from the terminal graph into , i.e. a vertex of , and similarly when it corresponds to an edge of . In other words, it is the category universally obtained by gluing together vertices of along edges of : the free category on . If we vary , fixing and varying for other choices of , we can obtain different left adjoint functors also. For : free category underlying graph free groupoid underlying graph of core discrete category clique graph on objects discrete category of connected components disconnected graph of objects We seek to universally determine the zigzag functor of definition in this manner. That is, due to proposition , should be obtained as the left Kan extension of some internal coalgebra in for some essentially algebraic theory , as in the following diagram: Nick Hu2024830https://forest.nickx.hu/coherent-inverses-4KT5/coherent-inverses-4KT5/coherent-inverses-4KT5/zigzag internal coalgebradefinition Define a functor , given on objects by each (augmented) simplex maps to a category with objects and morphisms for each ; each monotone map maps to a category generated by gluing together and , while adding morphisms when , and when , and quotienting together parallel morphisms; each composable pair of monotone maps maps to the category generated by gluing together , , and adding morphisms as above. Such a functor is immediately a -coalgebra internal to , as it preserves the colimits of corollary . Thus, by left Kan extension it determines a left adjoint functor , which we define to be . We now demonstrate that this indeed captures the same notion of zigzag category as in definition . Nick Hu2024830https://forest.nickx.hu/coherent-inverses-77KN/coherent-inverses-77KN/coherent-inverses-77KN/Explicit description of example Let . We can compute explicitly with a colimit formula where the comma category is the category of elements . That is, it is the category as the tip of the universal cocone, which has a leg for each object of given by where lives in the fibre of over ; , where is a morphism of living in the fibre of over ; where and are composable morphisms of , respectively living in the fibres of over and . By interpreting this colimit as a glueing construction over the categories in the image of , this is realised as the category with objects for each living in the fibre of over , morphisms for each morphism of living in the fibre of over , morphisms such that all four triangles in independently commute whenever they exist. This explicit description, along with the fact that , will play a larger role in section where we treat zigzags as graph-like data. Nick Hu2025523https://forest.nickx.hu/coherent-inverses-XXV2/coherent-inverses-XXV2/coherent-inverses-XXV2/The category example is an object of , and moreover every fibre over each object and morphism is trivial, so the description of is particularly simple. Objects are given by, for each , and each monotone map generates a collection of morphisms Now let be some zigzag category, regarded as an object of with the singular projection. A morphism is precisely a compatible family of choices for zigzags whose singular projection is , and zigzag maps whose singular projection is in . From the adjunction , there is a bijection of morphisms which states that, replacing with , this family is precisely a -labelling of the objects and morphisms of such that they assemble into zigzags and zigzag maps, which is exactly the compatibility condition. Nick Hu20241216https://forest.nickx.hu/coherent-inverses-I4ER/coherent-inverses-I4ER/coherent-inverses-I4ER/example revisitedremark Let be the terminal category. , as a right adjoint, preserves limits. Hence, is terminal in , so its image under the forgetful domain projection is , as the terminal object of is . In effect, this captures the same construction as definition — which encapsulates the idea that all the combinatorics of the zigzag category can be contained within one gadget — with more powerful tools than polynomial functors, which will be useful for an enriched generalisation of the zigzag polynomial functor. Indeed, it is not clear what an ‘enriched polynomial functor’ should be, but the zigzag internal coalgebra construction is more amenable to enrichment. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-zigzag-colimits/coherent-inverses-zigzag-colimits/coherent-inverses-zigzag-colimits/Colimits in the zigzag categorysection By casting the zigzag category abstractly as a parametric right adjoint, we can develop an alternative (but equivalent) perspective on colimits in the zigzag category as presented in [§ 3, high-level-methods-for-homotopy-construction-in-associative-n-categories]. We describe how to compute colimits in . Ultimately, this boils down to a kind of colimit procedure for . Every zigzag category admits a canonical singular projection functor , which takes a zigzag category to its underlying shape, and such a functor admits certain opcartesian lifts. Informally, given some diagram , computing the colimit of reduces to the two-step procedure: first, compute the underlying shape of the colimit of , by taking a colimit of the composite diagram , failing if this fails; use colimits in to fill in the shape with appropriate data to make this a universal cocone in . We perform the second step via the adjunction defining the zigzag functor. In order to explicate this, we view a colimit as a kind of pointwise left Kan extension. By doing so, we assert that any such colimit exists exactly when all the pointwise left Kan extensions exist. Nick Hu2024918https://forest.nickx.hu/coherent-inverses-97Q8/coherent-inverses-97Q8/coherent-inverses-97Q8/under cocone category of definition For a small category , the under cocone category is the category with a freely adjoined terminal object . Nick Hu2024918https://forest.nickx.hu/coherent-inverses-422E/coherent-inverses-422E/coherent-inverses-422E/Colimit as a pointwise left Kan extensionlemma[Proposition 6.3.10, category-theory-in-context] Let be a functor. The data of the colimit is equivalent to the pointwise left Kan extension along the inclusion : given by and for each unique morphism in . The inclusion is fully faithful, so the unit of the left Kan extension is an isomorphism, and in particular as is an inclusion can be chosen to be the identity. We introduce some 2-categorical aspects of slice categories over , which will be necessary to rewrite colimits in as colimits in , but without fully using 2-categorical language. Nick Hu2024918https://forest.nickx.hu/coherent-inverses-C2YY/coherent-inverses-C2YY/coherent-inverses-C2YY/slice Hom category definition With respect to the slice category , between any two objects and , the set of morphisms forms a (small) category with objects functors satisfying ; morphisms natural transformations . This is a special case of the 2-categorical notion of strict slice 2-category, over the strict 2-category . Due to this strictness, a morphism in this category is any natural transformation without any further conditions; this yields the following lemma. Nick Hu2024918https://forest.nickx.hu/coherent-inverses-EIW0/coherent-inverses-EIW0/coherent-inverses-EIW0/Forgetful domain functor locally fully faithfullemma For any category , there is a forgetful functor which sends . Moreover, for each slice Hom category , induces a fully faithful functor The interaction between this and a parametric right adjoint endofunctor on allows us to conduct a lifting of pointwise left Kan extensions. Nick Hu2024918https://forest.nickx.hu/coherent-inverses-VCOB/coherent-inverses-VCOB/coherent-inverses-VCOB/Lifting a pointwise left Kan extension through a parametric right adjoint endofunctor on theorem Let be a parametric right adjoint endofunctor, determined by: with the natural bijection of the adjunction denoted by For functors and in , where is an inclusion, let their projection onto domain via be denoted by and respectively. Then, we have a bijection of commuting diagrams in : and i.e. Nick Hu2024918proof Fix any arbitrary functor . By lemma , is in bijective correspondence with some via . Now calculate Because is a natural bijection, any functor can be written as . The result follows from the Yoneda lemma. Nick Hu2024921https://forest.nickx.hu/coherent-inverses-69G2/coherent-inverses-69G2/coherent-inverses-69G2/Colimits in are computed from colimits in corollary As the zigzag functor is a parametric right adjoint (definition ), left Kan extensions along inclusions in are computed as left Kan extensions along in via theorem . Moreover, because colimits are specific left Kan extensions (lemma ), all colimits in reduce to colimits in . In order to transport such an inclusion across the adjunction defining the zigzag functor, it is necessary to see as not merely a functor, but a morphism in ; that is, every object in its domain must live in some fibre over and must respect the singular projection. For the associated to the left Kan extension that computes a colimit, as in the definition , this means obtaining from some diagram (which is naturally a morphism of via singular projection) a way to enhance the canonical inclusion functor to be additionally an inclusion living over , as in: =10pt, ""'] \end {tikzcd} ]]> thus building a morphism of . This is tantamount to picking some over which the freely adjoined terminal object lives; to do so canonically, such that the left Kan extension of corollary computes the colimit of , we use the observation that the singular projection preserves connected colimits if admits a terminal object. Nick Hu2024923https://forest.nickx.hu/coherent-inverses-13T2/coherent-inverses-13T2/coherent-inverses-13T2/singular projection preserves connected colimitsproposition[Proposition 39, high-level-methods-for-homotopy-construction-in-associative-n-categories] Whenever admits a terminal object, then the singular projection functor preserves connected colimits. This justifies step 1 of the two-step procedure outlined at the beginning of this section. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-framed-zigzags-enriched/coherent-inverses-framed-zigzags-enriched/coherent-inverses-framed-zigzags-enriched/framed zigzag enriched categorysection We close this chapter by giving an enriched zigzag category construction, which provides a categorical foundation for the theory of coherently invertible generators in associative -categories. We describe the abstract universal properties of the enriched framed zigzag enriched category. We will set this up as a generalisation of definition , where is replaced by , the category of small -categories and -functors. First, we determine the essentially algebraic theory of as an analogue of proposition . For consistency, we will use an alternative notation to the ordinal notation from before, e.g. writing for . Nick Hu2024830https://forest.nickx.hu/coherent-inverses-X94B/coherent-inverses-X94B/coherent-inverses-X94B/dense subcategory of proposition Determine a subcategory of given by -categories and -functors as follows: , bend left] (a) edge (b); \draw [->, bend right] (a) edge (b); \end {tikzpicture} \} \ar [r, shift left] \ar [r, shift right] & {[\wedge ]} = \{ \begin {tikzpicture} \node [draw=none] (a) at (0,0) {$\cdot $}; \node [draw=none] (b) at (1,0) {$\cdot $}; \node [draw=none, anchor=south] (leq) at (0.5,0) {$\Uparrow $}; \node [draw=none, anchor=north] (leq) at (0.5,0.175) {$\Downarrow $}; \draw [->] (a) edge (b); \draw [->, bend left=60] (a) edge (b); \draw [->, bend right=60] (a) edge (b); \end {tikzpicture} \} \end {tikzcd} ]]> where the top part of the diagram is analogous to proposition , and every remaining morphism (each of which is a -functor) is a non-degenerate inclusion. Such a subcategory inclusion is dense. Nick Hu2024830proof A -functor is a more restrictive version of a functor in the sense that it not only must preserve identity and composition, but also ordering and meets in Hom -semilattices (doing so as property, not structure — there is no choice in the way such properties are preserved; the -functor either preserves them or fails to exist by virtue of not preserving them). Thus, as with a functor, it is determined entirely by its action on objects and morphisms. Drawing on the proof of proposition , the presence of objects and ensures that the relative nerve is faithful. To see that it is also full, the presence of , along with the inclusions into it from , allows the relative nerve to detect the ordering of each Hom -semilattice; similarly, allows the meet operation itself to be detected. The inclusions of say that this meet operation is with respect to the ordering of the Hom -semilattices. Note that if we were instead working in a 2-categorical setting as opposed to our -enriched setting, where all the 2-cell structure is thin, we would need many more morphisms to ensure additional properties are preserved, e.g. a 2-functor preserves identity 2-cells. In the present case, for an -functor this is automatic from monotonicity with respect to Hom -semilattices. In order to repeat the derivation of the functor, we now need to find its image at the terminal , analogously to remark . To do this, we make use of the adjunction corollary . Nick Hu20241216https://forest.nickx.hu/coherent-inverses-L7A7/coherent-inverses-L7A7/coherent-inverses-L7A7/-enriched definition Define the -category (definition ) as follows: objects finite ordinals for ; morphisms non-empty sets of monotone maps ; composition is given by the set of all compositions, with identities given by singletons containing the identity monotone map; meet set union. Note that (as with every -category) the ordering in each Hom -semilattice is determined by ; in this case, a set of monotone maps is below another exactly when it is a superset. This -category is obtained by taking the ordinary category , freely turning it into a -category via in lemma , and then freely turning that into an -category along corollary . Nick Hu20241216https://forest.nickx.hu/coherent-inverses-OXIE/coherent-inverses-OXIE/coherent-inverses-OXIE/dense subcategory of proposition Let denote the dense subcategory of as in proposition . Then, by proposition , we have a dense subcategory of given by the category of elements . This category has objects pairs , where and is a -functor. Analogously to proposition , this is a version of proposition but indexed by objects of . That is, objects are one of five types (corresponding to the value of ): objects of : (possibly empty) ordinals ; morphisms of : non-empty sets of monotone maps ; composable pairs of morphisms of : pairs of non-empty sets of monotone maps ; witnesses to the ordering relation on Hom -semilattices of : a non-empty set of monotone maps equipped with a non-empty subset; witnesses to the meet operation on Hom -semilattices of : two non-empty sets of monotone maps whose union is (necessarily) non-empty. Let denote the dense full subcategory of which this induces. Next, we present a sequence of lemmas that use density presentations to characterise left adjoints (as in proposition ). Nick Hu20241216https://forest.nickx.hu/coherent-inverses-BS1R/coherent-inverses-BS1R/coherent-inverses-BS1R/density presentation of lemma The dense full subcategory inclusion of admits a density presentation given by the diagrams Nick Hu20241216https://forest.nickx.hu/coherent-inverses-U5WG/coherent-inverses-U5WG/coherent-inverses-U5WG/-coalgebracorollary An internal -coalgebra in a category is precisely a functor which preserves the colimits Nick Hu20241216https://forest.nickx.hu/coherent-inverses-6QGR/coherent-inverses-6QGR/coherent-inverses-6QGR/density presentation of corollary The dense full subcategory inclusion of admits a density presentation given by diagrams of the form Nick Hu20241216https://forest.nickx.hu/coherent-inverses-VBSW/coherent-inverses-VBSW/coherent-inverses-VBSW/-coalgebracorollary An internal -coalgebra in a category is precisely a functor which preserves the colimits Nick Hu20241217https://forest.nickx.hu/coherent-inverses-6E0J/coherent-inverses-6E0J/coherent-inverses-6E0J/framed zigzag internal coalgebradefinition Define a functor , given on objects by each (augmented) simplex maps to a -category with objects and morphisms for each , where the ordering and meets of morphisms are determined trivially; each non-empty set of monotone maps maps to a -category generated by, for each , gluing together and , while adding morphisms when , and when , and the morphisms obtained from adding 2-cell fillers which is then quotiented along each copy of and ; each composable pair of non-empty sets of monotone maps maps to the -category generated by gluing together , , and adding morphisms and 2-cell fillers as above; for a non-empty set of monotone maps equipped with a non-empty subset , the -category generated by and , glued along and as above, with 2-cell fillers from component-wise for every to its corresponding instance ; for non-empty sets of monotone maps (whose union is non-empty: ), the -category generated by , , and , glued along and as above, with 2-cell fillers from component-wise for every and to its corresponding instance . This functor is a -coalgebra internal to , because it preserves the colimits of corollary . Analogously to definition , by left Kan extension it determines a left adjoint functor , which we define to be -enriched version of . This is automatically a left adjoint functor, so we take its right adjoint to be the -enriched version of , which we call the framed zigzag functor. Nick Hu20241217https://forest.nickx.hu/coherent-inverses-51XQ/coherent-inverses-51XQ/coherent-inverses-51XQ/framed zigzag functortheorem We obtain a framed zigzag functor as a parametric right adjoint given by Finally, we give a concrete description of the framed zigzag enriched category, similar to definition . Nick Hu2024830https://forest.nickx.hu/coherent-inverses-XQVM/coherent-inverses-XQVM/coherent-inverses-XQVM/framed zigzag enriched categorydefinition For a -category , we define the framed zigzag enriched category as the category with: objects (framed zigzags) iterated cospans of ; e.g. for objects , and morphisms of for , morphisms (framed zigzag maps) For a fixed pair of iterated cospans , consider collections of morphisms of which arrange into a (singular-singular) monotone map of singular levels, combined with a dually determined antiparallel (regular-regular) monotone map of regular levels, along with ‘composite’ (regular-singular) maps with 2-cell fillers, such that the induced planar diagram in exists. E.g. diagrams in of the form: where the morphisms labelled arrange into the singular monotone map , and the morphisms labelled arrange into the (antiparallel) regular monotone map , and moreover these maps induce ‘composite’ morphisms labelled with associated 2-cell fillers. Define a partial order on these collections exactly when and have the same underlying shape (given by underlying singular/regular monotone maps), and in for all valid . A framed zigzag map is given by a non-empty downwards-closed (with respect to ) set of these collections. meets set unions. Composition of framed zigzag maps is given by taking meets of all possible compositions of along composition of monotone maps (as in ), as follows. For each , compose their underlying singular monotone maps as monotone maps to determine the underlying singular monotone for the pair in the composite; this suffices to determine the shape of the composition at the pair (i.e. the dual regular monotone map, and ‘composite’ maps); then, use the structure to determine its -labelling on each component: for each valid , where is some regular or singular level of and is some regular or singular level of , define its -labelling to be the meet of every composite of components of followed by with type . For example: Here, the component of the composite is the meet of the 7 possible paths from to in the left diagram, which simplifies to the meet of the 3 paths as above — a priori, none of the three maps is below another. Note that singular-singular and regular-regular components are always determined by a unique path, but not regular-singular components. The composite is then given by the downwards closure of the resulting set with respect to . This set is non-empty by construction, as and are. The identity framed zigzag map is given by the principal downwards-closed set generated by identity morphisms of equipped with the identity monotone map. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-collapse/coherent-inverses-collapse/coherent-inverses-collapse/Collapsing framed zigzagschapter In this chapter, we explore the more algorithmic aspects of the theory of framed zigzag enriched categories, and in particular the contraction algorithm, which is used to construct complex zigzag homotopies. Fundamentally, an object of an iterated framed zigzag enriched category is a structure equipped with ‘slicing’ and ‘explosion’ operations (see explosion bundle), which we represent by certain forms of (enriched) functors. These can be represented graph-theoretically as -graphs, which are then amenable to computer implementation. Our algorithms centre on a notion of collapse for -graphs — in a way that will be made precise, this procedure replaces such a functor by a simpler one, in a way that preserves ‘colimits’. This algorithm serves as a building block for the key functionality of homotopy.io, featuring in the construction of algebraic signatures via the generator creation algorithm, homotopy construction via contraction algorithm for framed zigzags, and a typechecking algorithm which determines the validity of diagrams. We argue for correctness of this algorithm by determining its behaviour as the computation of a special kind of colimit, which we extend to an argument for correctness of the contraction algorithm by showing that it admits certain well-behavedness conditions. Although the framed zigzag construction is -enriched, we will consider the more general case of -enrichment. Informally, this is because the structure in each Hom semilattice is only used as a technical trick to ensure that the framed zigzag enriched category has composition. It is in fact the case none of the operations we wish to consider make use of composition of morphisms anyway, which also influences our decision to consider -graphs instead ​ . We can justify this change in setting by transparently transporting across corollary . Namely, this implies that everything we say about colimits in -categories will also apply to -categories. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-colimits/coherent-inverses-colimits/coherent-inverses-colimits/Colimit theorysection We begin in the general setting by considering the new colimit-type constructions available in the enriched setting of -categories. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-oplax/coherent-inverses-oplax/coherent-inverses-oplax/Oplax category theorysection Because our setting is (at least) -enriched, it is possible to express certain notions of category theory that are more ‘lax’, i.e. some commutativity conditions are replaced by the existence of 2-cell fillers, not necessarily in both directions. One such notion is that of an oplax colimit, which is a relaxed version of a colimit. We first recall some basic facts about oplax colimits ​ in the -enriched setting. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-UKT4/coherent-inverses-UKT4/coherent-inverses-UKT4/lax natural transformationdefinition A lax natural transformation between -functors is: for each , a collection of component 1-morphisms in ; for each morphism in , a 2-morphism filler for in : The collection of all lax natural transformations is the poset denoted by , which is ordered component-wise. Nick Hu2023125https://forest.nickx.hu/coherent-inverses-W0JK/coherent-inverses-W0JK/coherent-inverses-W0JK/oplax coconedefinition An oplax cocone over is a lax natural transformation . That is, a family of 1-morphisms in , , such that for each morphism in , there is a 2-morphism filler : Note that an oplax cocone is given by a lax natural transformation. Oplax cocones over a diagram with a fixed tip are partially ordered pointwise, and form a poset . Nick Hu2023125https://forest.nickx.hu/coherent-inverses-QTSK/coherent-inverses-QTSK/coherent-inverses-QTSK/oplax colimitdefinition The oplax colimit of is the universal oplax cocone : for any other oplax cocone , there exists a unique 1-morphism in such that for all , : Also, for any other oplax cocone over with the same tip, , which factors via , as above, and additionally is pointwise-ordered above (for all , the 2-morphism filler exists), then , and vice versa. In other words, the universal property of the oplax colimit is equivalently expressed as a natural order isomorphism of posets: Nick Hu2023125https://forest.nickx.hu/coherent-inverses-TIQR/coherent-inverses-TIQR/coherent-inverses-TIQR/oplax colimits are unique up to unique isomorphismlemma The oplax colimit of a -functor is unique up to unique isomorphism. Nick Hu2023125proof Let and be oplax colimits over a diagram . Through this, we obtain unique factorisations and : also factors through itself via the identity: Now, the first triangle can be pasted against itself, and the uniqueness of the factorisation allows us to deduce , and symmetrically . Nick Hu2023125proof We present an alternative and more abstract proof. Let also have the same universal property as : then deduce that , and hence by the -enriched Yoneda lemma, . Nick Hu2023123https://forest.nickx.hu/coherent-inverses-oplax-pathologies/coherent-inverses-oplax-pathologies/coherent-inverses-oplax-pathologies/Pathological examplessection An oplax cocone is a relaxed version of a cocone, and in general there may exist more oplax cocones over any diagram of -categories (each cocone is trivially also an oplax cocone). This leads to some behaviours that are not present in the theory of colimits in the unenriched setting, which we detail in this section. The property of having an oplax colimit is not stable under extending a diagram by identities. Nick Hu2024530https://forest.nickx.hu/coherent-inverses-Q4FX/coherent-inverses-Q4FX/coherent-inverses-Q4FX/oplax colimits unstable under extension by identityexample Let be the -category generated by: and and given respectively by diagrams: has both an oplax colimit and a conical colimit, given by the identity cocone with tip and leg given by . However, the identity cocone over is not an oplax colimit: the oplax cocone =10] \end {tikzcd} ]]> exists; if the identity cocone were universal, then this oplax cocone would factor through it, which is tantamount to requiring a morphism which is simultaneously equal to and , which does not exist. In ordinary category theory, any diagram that admits a terminal object in its indexing category admits as a colimit. This is not so for oplax colimits, as the following example shows. Nick Hu2024530https://forest.nickx.hu/coherent-inverses-J1Q9/coherent-inverses-J1Q9/coherent-inverses-J1Q9/oplax colimits not given by terminal indexexample Let be the walking arrow, be the -category generated by: and and given by the diagrams which choose the morphisms and respectively. admits only one oplax cocone, with tip and legs and , and so this is an oplax colimit. However, admits two oplax cocones: =10] \end {tikzcd} ]]> =10] \end {tikzcd} ]]> and neither of these factor through each other, so the oplax colimit does not exist. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-weighted/coherent-inverses-weighted/coherent-inverses-weighted/Oplax, marked, and weighted colimitssection The notion of oplax colimit is a specific instance of a more general concept: a weighted colimit. Nick Hu202493https://forest.nickx.hu/coherent-inverses-KV1B/coherent-inverses-KV1B/coherent-inverses-KV1B/weighted colimitdefinition Given a -functor and additionally a -functor , called a weight, the weighted colimit of with respect to is an object with the universal property in . In enriched category theory, it becomes the case that the ‘conical’ shape of an ordinary colimit is insufficiently expressive — for instance, a -category may possess all conical colimits but not all the oplax colimits. The weight associated to a weighted colimit generalises this conical shape, and is the appropriate notion of colimit in this setting. Nick Hu202493https://forest.nickx.hu/coherent-inverses-CSVT/coherent-inverses-CSVT/coherent-inverses-CSVT/weighted colimits as generalised coconesremark A cocone in ordinary category theory is expressed by a natural transformation into a constant functor. For instance, for some diagram , a cocone with tip is given by the data of some . In the context of -categories, these cocones arrange themselves into the poset . For the colimit to possess the universal property means that every cocone with tip is in (order-preserving) bijection with a morphism of , and this is precisely what it means for to be the tip of a universal cocone — that every cocone factors uniquely through the one with tip . We can observe further, letting denote the terminal poset, that there is an isomorphism witnessed by the map where is given component-wise as . A weighted colimit is a colimit of generalised shape, replacing with any . Explicitly, at each object , there is a poset of ‘legs’ emanating from , valued in the Hom poset , packaged as a natural transformation . It was first observed by [Theorem 11, limits-indexed-by-category-valued-2-functors], in the 2-categorical case, that an oplax colimit is induced as a weighted colimit for a specific weight; here, we refine the notion for -categories. Nick Hu2024729https://forest.nickx.hu/coherent-inverses-654O/coherent-inverses-654O/coherent-inverses-654O/oplax weightdefinition Let be a -functor. The oplax colimit of is equivalently given by the weighted colimit where the weight sends each to the set of morphisms in with domain , ordered by whenever there exists some -morphism which oplaxly extends to : which is something like the oplax coslice as a poset. Each -morphism is sent to the monotone map of precomposition by , which preserves identity, composition, and is itself monotone. Nick Hu202495https://forest.nickx.hu/coherent-inverses-RQWC/coherent-inverses-RQWC/coherent-inverses-RQWC/example via weighted colimitsexample Assuming the same setup as example , deduce the oplax weight as the -functor whose image in is Via remark , this says that a generalised cocone with respect to this weight for the diagram with tip is one which has two legs and over and one leg over , and moreover the top leg must be . Pictorially, this looks like which is exactly what it means for and to form an oplax cocone over . Nick Hu202495https://forest.nickx.hu/coherent-inverses-6J1T/coherent-inverses-6J1T/coherent-inverses-6J1T/oplax colimit is a weighted colimitlemma[Theorem 11, limits-indexed-by-category-valued-2-functors] For a -functor , whenever it exists, its oplax colimit is given by a weighted colimit with respect to the weight as in definition : With this understanding of oplax colimit, we seek to refine the weight to find some intermediate notion of weighted colimit that interpolates between an ordinary (conical) colimit (which is very strict) and an oplax colimit (which is very lax). To do so, we introduce the notion of marked colimit. Nick Hu202495https://forest.nickx.hu/coherent-inverses-5VBZ/coherent-inverses-5VBZ/coherent-inverses-5VBZ/markingdefinition Given a -functor , a marking is a class of morphisms of which includes all of the identities and is closed under composition. Nick Hu202495https://forest.nickx.hu/coherent-inverses-C2BO/coherent-inverses-C2BO/coherent-inverses-C2BO/marked weightdefinition The marked weight associated to a -functor equipped with a marking is defined similarly to the oplax weight (definition ), except each poset in its image is additionally localised at The fact that is closed under composition ensures that is well-defined as a -functor (on morphisms). Recall that each poset in the image of the weight for a weighted colimit specifies how many distinct legs live over any particular ; this localisation strictifies the oplax weight by forcing some of those legs to be identified — namely, those that are connected by marked morphisms. Nick Hu202495https://forest.nickx.hu/coherent-inverses-S9V9/coherent-inverses-S9V9/coherent-inverses-S9V9/marked colimitdefinition The marked colimit of a -functor equipped with a marking , when it exists, is given by the weighted colimit with respect to the marked weight . It is clear that when the marking is minimal (i.e. containing only identity morphisms), the notion of marked colimit coincides exactly with oplax colimit. Moreover, when the marking is maximal (containing all possible morphisms), then a marked colimit is a colimit, as the following example illustrates. Nick Hu202496https://forest.nickx.hu/coherent-inverses-SHM3/coherent-inverses-SHM3/coherent-inverses-SHM3/example with marked colimitsexample We will re-examine example in the context of marked colimits. Recall that the reason that the diagram did not admit an oplax colimit was due to the presence of too many oplax cocones. If instead we equip to the marking which selects all morphisms of , then the marked weight becomes the -functor which is constant at the point; i.e., the generalised cocones associated to this weight are ordinary cocones, and the marked colimit in this case coincides with the colimit (as in remark ), which we obtain as equipped with identity morphisms for legs. We can further use remark to develop an explicit description of marked colimit. Nick Hu2024917https://forest.nickx.hu/coherent-inverses-5128/coherent-inverses-5128/coherent-inverses-5128/marked coconedefinition Given a -functor equipped with a marking , a marked cocone over is given by a lax natural transformation where for each in , the lax naturality square =10pt, "\sim "'] \end {tikzcd} ]]> is required to commute strictly (i.e. it inverts). Equivalently, this is an oplax cocone where some of the 2-cell fillers are required to be invertible, as determined by the marking . We denote the poset of marked cocones between two -functors and as . Nick Hu2024917https://forest.nickx.hu/coherent-inverses-N8XL/coherent-inverses-N8XL/coherent-inverses-N8XL/marked colimit is a universal marked coconeproposition A -functor equipped with a marking has a marked colimit if and only if it admits a universal marked cocone; that is, a marked cocone such that any other marked cocone factors through uniquely by some morphism Succinctly, is an object of such that, for any other object , there is an order isomorphism of Hom posets: Nick Hu2024917proof It suffices to show that there is an order isomorphism Suppose that we have some marked cocone ; we can define a natural transformation componentwise by If is a marked morphism in , then it is identified with in the poset . is natural because , and is a -functor. is injective, and an order embedding because the poset is ordered pointwise. Now instead suppose that we have some natural transformation . Define componentwise by . We need to show that is a marked cocone with respect to . That is, =10pt, ""'] \end {tikzcd} ]]> for each in , where the 2-cell filler inverts if is marked by . The naturality of determines an equality of morphisms also, is a monotone map, so because , deduce that combining these, we derive the 2-cell filler which inverts when is marked as required. A simple calculation shows that is inverse to . Thus, we have established that is an invertible order embedding, and hence an order isomorphism. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-graded/coherent-inverses-graded/coherent-inverses-graded/Colimit theory of graded -categoriessection In order to deal with the pathologies outlined in example and example , we restrict our attention to marked colimits (recall that marked colimits subsume both colimits and oplax colimits) in particularly well-behaved -categories. Nick Hu2024116https://forest.nickx.hu/coherent-inverses-L0IM/coherent-inverses-L0IM/coherent-inverses-L0IM/Finite direct -categorydefinition A finite -category is direct whenever it satisfies any of the following equivalent conditions: there is no infinite sequence of composable non-identity morphisms in ; an antiparallel pair of morphisms only exists when both are identity morphisms and ; there is a function on the set of objects of such that every morphism in increases dimension (non-identities strictly so); is skeletal, and every endomorphism is necessarily the identity. Weighted colimits in direct -categories are uniquely determined on-the-nose (as opposed to up-to-unique-isomorphism), as every direct category is gaunt, which implies that marked colimits are too. We note that must be given as structure, as it is not uniquely determined (for instance, given one valid choice for , another one can be generated by composition with the successor function on natural numbers). However, there is a canonical choice for . Firstly, observe that finite direct -categories induce a graded poset on their set of objects. Nick Hu2024116https://forest.nickx.hu/coherent-inverses-5YGZ/coherent-inverses-5YGZ/coherent-inverses-5YGZ/underlying (graded) poset of a finite direct -categorydefinition Let be a finite direct -category. The objects of form an underlying (graded) poset by whenever there exists a morphism in . This is precisely the posetal reflection of the image of in the adjoint functor of lemma . Then, we extend this as follows. Nick Hu2025522https://forest.nickx.hu/coherent-inverses-L45G/coherent-inverses-L45G/coherent-inverses-L45G/graded -categorydefinition A graded -category is a finite direct -category equipped with the dimension function determined by the grading of definition . In other words, the dimension of any is given by the length of the longest maximal chain of composable non-identity morphisms ending at . In this case, we refer to the as the rank of . Marked colimits behave more simply in graded -categories; for instance, we can make the following observations. Nick Hu2024116https://forest.nickx.hu/coherent-inverses-PB63/coherent-inverses-PB63/coherent-inverses-PB63/Rank of marked coconeslemma Let be a -functor where is graded. Any marked cocone over has a tip with rank at least . Nick Hu2024116proof Let be such an marked cocone, for . For any , we have a leg of the cocone, a morphism . Hence, . There is a special class of -functors that have yet simpler marked colimits. Nick Hu202519https://forest.nickx.hu/coherent-inverses-4G17/coherent-inverses-4G17/coherent-inverses-4G17/directly surjective -functordefinition A -functor is directly surjective if is graded, and is surjective on max-rank objects of . In general, a marked colimit is given by a universal marked cocone (proposition ); in this case, we have restricted our setting to be such that at most one such marked cocone can exist and must be in the image of its diagram . Nick Hu2024116https://forest.nickx.hu/coherent-inverses-OBOY/coherent-inverses-OBOY/coherent-inverses-OBOY/marked cocones over directly surjective -functorsproposition Let be a directly surjective -functor. Any marked cocone over has tip given by the element of with maximal rank, which is necessarily unique among maximal rank elements of . Nick Hu2024116proof Combine lemma with the fact that such an marked cocone cannot have strictly higher rank than every element of , because is directly surjective. Moreover, if are both maximal rank, then no marked cocone can exist: by assumption, there are objects of such that and , but both legs of the marked cocone must be the identity, which is a contradiction. Necessarily, this is a marked colimit when it exists. In fact, it is so exactly if there are no other marked cocones. Nick Hu20241114https://forest.nickx.hu/coherent-inverses-M1PP/coherent-inverses-M1PP/coherent-inverses-M1PP/marked colimits of directly surjective -functorscorollary Let be a directly surjective -functor. The marked colimit of exists if and only if there is exactly one marked cocone over . Nick Hu20241114proof It is clear to see that if there is exactly one marked cocone over , then it must be an marked colimit. Otherwise, assume the existence of such an marked colimit. proposition implies that every marked cocone over has the same tip, so the universal factoring map must be an endomorphism. But because is direct, it must moreover be the identity. This implies that every marked cocone over is equal. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-collapse-contraction-typechecking/coherent-inverses-collapse-contraction-typechecking/coherent-inverses-collapse-contraction-typechecking/Collapse, contraction, and typecheckingsection In this section, we describe the concrete algorithmic aspects of the framed zigzag construction, which relate framed zigzags to combinatorial encodings of string diagrams. First, we introduce a notion of collapsibility, which will be principal to our development. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-collapsible/coherent-inverses-collapsible/coherent-inverses-collapsible/Collapsible morphisms with respect to a diagramsection In ordinary category theory, one often considers certain functors over a fixed codomain whose domain represents some kind of ‘shape’. For instance, a commutative diagram in a category is formally some functor , where is usually some small (finite) category specified freely over some -graph, and moreover thin (this is what it means for such a ‘diagram’ to commute). When (op)laxity is concerned, some care is required in the treatment of a ‘diagram’ (as opposed to a ‘functor’). Nick Hu20241030https://forest.nickx.hu/coherent-inverses-2WYC/coherent-inverses-2WYC/coherent-inverses-2WYC/atomdefinition[Definition 2.1, discrete-morse-theory-and-localization] Let be a -category. A morphism of is an atom if is minimal: ; for any oplaxly commuting triangle in : at least one of the outer faces is the identity morphism. In the case that both are, then is an identity morphism necessarily. That is, is oplaxly indecomposable. Nick Hu20241030https://forest.nickx.hu/coherent-inverses-4O2T/coherent-inverses-4O2T/coherent-inverses-4O2T/oplax diagramdefinition An oplax diagram is a -functor of the form , where is freely generated by some poset seen as a thin -category as follows: every Hom poset is a poset of directed paths of morphisms of which connect to , ordered by whenever is a path which can be obtained by deleting vertices of , admitting the direct path as the minimal element. Nick Hu20241212https://forest.nickx.hu/coherent-inverses-RRZ9/coherent-inverses-RRZ9/coherent-inverses-RRZ9/simple oplax diagramdefinition An oplax diagram is simple when its underlying -graph is simple. Nick Hu20241028https://forest.nickx.hu/coherent-inverses-NBLG/coherent-inverses-NBLG/coherent-inverses-NBLG/collapsible morphismdefinition Let be an oplax diagram. is weakly collapsible if is an atom; sends to an identity morphism. If, additionally, every 2-cell filler which mentions is sent by to an identity 2-cell filler, then is collapsible. Write if and are connected by collapsible morphisms. These definitions allow us to isolate a subclass of the morphisms in the index of an oplax diagram which map to identities. Nick Hu20241031https://forest.nickx.hu/coherent-inverses-L54I/coherent-inverses-L54I/coherent-inverses-L54I/Some collapsible morphismsexample Consider the following simple oplax diagram: \j; } } } , graphs/math nodes , graphs/nodes={circle, fill=white, inner sep=1pt} , subgraph text none, ] \path graph[spring layout]{ 0 / "x_0"; 1 / "x_1"; 2 / "x_2."; 3 / "y_3"; 0 ->["$ \text {id}_{x} $"{above right}, green] 1 ->["$ \text {id}_{x} $"', ""{name=BR}, orange] 2; 0 ->["$ \text {id}_{x} $", ""'{name=TR}] 2; 0 ->["$g$"', ""{name=TL}] 3 <-["$f$"', ""{name=BL}] 1; }; \begin {scope}[on background layer] \path (TL) edge[commutative diagrams/Rightarrow, shorten >=2pt] (1) (TR) edge[commutative diagrams/Rightarrow, shorten >=2pt] (1); \end {scope} \end {tikzpicture} ]]> From this diagram, we can deduce a few things: the morphism is not an atom, as it oplaxly decomposes into ; thus, it is neither weakly collapsible nor collapsible; the morphism is weakly collapsible, being an atom which is labelled by an identity morphism; however, it is not collapsible, as the 2-cell filler is mapped to a non-identity 2-cell filler (given that ); the morphism is collapsible. We wish to find a ‘simplification’ procedure for simple oplax diagrams that preserves ‘colimits’, by ‘collapsing away’ collapsible edges, such that the end result is also a simple oplax diagram. At a glance, it may seem that the notion identifying objects along whenever there exists a collapsible morphism may suffice (weakly collapsible is not enough, as example shows: such a ‘collapse’ identifying and into some combined object must then admit two morphisms with codomain , with types and ). However, this turns out to be insufficient, as example will show. Instead, it becomes necessary to use extra structure associated to an oplax diagram, which we detail in the following section. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-strongly-collapsible/coherent-inverses-strongly-collapsible/coherent-inverses-strongly-collapsible/Strongly collapsible morphisms and well-formednesssection Here, the key idea is that we want to specify some extra data which allows us to express a notion of well-formedness. Nick Hu20241118https://forest.nickx.hu/coherent-inverses-QPQ4/coherent-inverses-QPQ4/coherent-inverses-QPQ4/tree-cover oplax diagramdefinition An oplax diagram is a tree-cover oplax diagram when it is equipped with a family of subsets of such that ; for all objects , ; for every , and do not overlap, which means either they are disjoint or one is a subset of another. Intuitively, this means that is a family of subsets of whose union is , for which the Hasse diagram under the inclusion ordering looks like a tree. For any , we denote by the oplax diagram given by restriction of to the full sub--category of whose domain only has objects in . Every morphism of admits a ‘smallest’ diagram in which it appears, where Nick Hu20241118https://forest.nickx.hu/coherent-inverses-ZCMQ/coherent-inverses-ZCMQ/coherent-inverses-ZCMQ/well-formed tree-cover oplax diagramdefinition A tree-cover oplax diagram is well-formed if for , if is a collapsible morphism of then it is a collapsible morphism of . Note that, in general, the operation of restricting an oplax diagram preserves collapsible morphisms, so the converse of this condition is automatic. This definition says that respects collapsible morphisms in the following sense: consider some maximal chain ; every morphism of will appear for the first time in the oplax diagram for some in this chain, at which point it remains present in the oplax diagrams for subsequent elements; this condition says that a morphism is collapsible in if and only if it is in the restricted oplax diagram in which it first appears. In other words, the global property of a morphism of being collapsible is compatible with the local property of being collapsible in some . Nick Hu20241121https://forest.nickx.hu/coherent-inverses-J2IZ/coherent-inverses-J2IZ/coherent-inverses-J2IZ/Non-well-formed tree-cover oplax diagramexample Consider the following tree-cover oplax diagram: [orange] x <-[orange] x}; { [name=s0, fresh nodes] x ->["$\gamma $"{name=c}] c <-["$\delta $"{name=d}] x}; { [name=r0, fresh nodes, nodes={xshift=1.5cm}] x / "x,"}; s0 x <- {r0 x, r1 x}; r0 x ->[gray, "$\alpha $"{name=a}] s0 c; s0 x' <- {r0 x, r1 x''}; r1 x ->[gray, "$\beta _0$"{name=b0}] s0 c; r1 x' ->["$\beta _1$"{name=b1}] s0 c; r1 x'' ->[gray, "$\beta _2$"{name=b2}] s0 c; }; \draw [/tikz/commutative diagrams/Rightarrow] (a) to (c); \draw [/tikz/commutative diagrams/Rightarrow] (a) to (d); \draw [/tikz/commutative diagrams/Rightarrow] (b0) to (s0 x); \draw [/tikz/commutative diagrams/Rightarrow] (b0) to (r1 x'); \draw [/tikz/commutative diagrams/Rightarrow] (b2) to (s0 x'); \draw [/tikz/commutative diagrams/Rightarrow] (b2) to (r1 x'); \node [draw, rounded corners, fit={(s0 x) (s0 x')}] {}; \node [draw, rounded corners, fit={(r1 x) (r1 x'')}] {}; \end {tikzpicture} ]]> where the unlabelled morphisms are identities and the boxes indicate the non-trivial parts of the tree cover. Assuming that , , and are not all equal morphisms, this diagram is not well-formed: each morphism in the top row is collapsible according to its minimal (which corresponds to the subdiagram contained within the topmost box), however in this instance they would not be collapsible according to the whole diagram. Nick Hu20241121https://forest.nickx.hu/coherent-inverses-1EX1/coherent-inverses-1EX1/coherent-inverses-1EX1/strongly collapsible morphismdefinition Fix a well-formed tree-cover oplax diagram . We say that a collapsible morphism is strongly collapsible (with respect to ) when for all such that and : any weakly collapsible morphism is collapsible, for any , if there exists a cospan then . Write when and are connected by strongly collapsible morphisms. Nick Hu20241214https://forest.nickx.hu/coherent-inverses-HSJ2/coherent-inverses-HSJ2/coherent-inverses-HSJ2/strong collapsedefinition In the setting of definition , let be the closure of the equivalence relation generated by strongly collapsible morphisms under composition. Define a sub--category by objects such that ; morphisms such that ; 2-cell fillers such that and . It can be readily checked that is a -category. Now, observe that the coequaliser (which exists because is cocomplete) admits the following universal property: is the -functor which sends each object to its equivalence class , and we call the strong collapse of . Nick Hu202512https://forest.nickx.hu/coherent-inverses-RF9D/coherent-inverses-RF9D/coherent-inverses-RF9D/weak collapsedefinition Analogously to definition , we define the weak collapse for any oplax diagram to be the quotient under connectivity by weakly collapsible morphisms. This is a larger quotient than that of strong collapse, i.e. possibly more objects may be identified as every strongly collapsible morphism is also weakly collapsible. Nick Hu20241118https://forest.nickx.hu/coherent-inverses-4NZH/coherent-inverses-4NZH/coherent-inverses-4NZH/Trivial tree coversremark Every oplax diagram can be equipped with a trivial tree cover given by Any tree-cover oplax diagram with given trivially as such is well-formed, and in this instance every collapsible morphism is strongly collapsible. Nick Hu20241121https://forest.nickx.hu/coherent-inverses-VR66/coherent-inverses-VR66/coherent-inverses-VR66/weakly collapsible, collapsible, and strongly collapsible morphismsexample Below we illustrate a well-formed tree-cover oplax diagram: [orange, postaction={decorate}] x <-[orange, postaction={decorate}] x}; { [name=s0, fresh nodes] x ->["$\gamma $"{name=c}] s <-["$\delta $"{name=d}] x ->[orange, postaction={decorate}] x <-[orange, postaction={decorate}] x}; { [name=r0, fresh nodes, nodes={xshift=3cm}] x [xshift=-1.5cm] ->[orange, postaction={decorate}] x <-[orange, postaction={decorate}] x}; s0 x <-[green] {r0 x, r1 x}; r0 x ->[gray, "$\alpha $"{name=a}] s0 s <-[gray, "$\beta $"{name=b}] r1 x; s0 x' <-[green] {r0 x, r1 x}; s0 x'' <-[orange] {r0 x', r1 x'}; s0 x'' <-[gray] {r0 x, r1 x, r0 x'', r1 x''}; s0 x''' <-[orange] {r0 x'', r1 x''}; }; \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (a) to (c); \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (a) to (d); \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (b) to (c); \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (b) to (d); \node [draw, rounded corners, fit={(r0 x) (r0 x'')}] {}; \node [draw, rounded corners, fit={(s0 x) (s0 x''')}] {}; \node [draw, rounded corners, fit={(r1 x) (r1 x'')}] {}; \node [right=5pt of r0 x''.south east] {,}; \end {tikzpicture} ]]> where unlabelled morphisms are the identity morphism ; , , , and are assumed to be distinct morphisms in the Hom poset , and the boxes indicate the non-trivial parts of the tree cover. This diagram has all three of weakly collapsible morphisms The weakly collapsible that are not collapsible are drawn in green. They are directly adjacent to non-trivial 2-cell fillers. collapsible morphisms The collapsible morphisms that are not strongly collapsible are drawn in orange (without crosses). They are collapsible, as they only participate in trivial 2-cell fillers, however they are not strongly collapsible because they connect disjoint parts of the tree cover by morphisms which are weakly collapsible but not collapsible. For example, consider the bottom-right orange collapsible morphism: the minimal which includes this morphism corresponds to the whole diagram; moreover, the endpoints of this morphism can be shifted along collapsible (orange) morphisms to admit a green weakly collapsible morphism, on the face of the 2-simplex . strongly collapsible morphisms × The strongly collapsible morphisms are drawn in orange, and additionally crossed out in red. Its strong collapse is as follows: ["$\gamma $"{name=c}] s <-["$\delta $"{name=d}] x}; { [name=r0, fresh nodes, nodes={xshift=1.5cm}] x / "x,"}; s0 x <- {r0 x, r1 x}; r0 x ->[gray, "$\alpha $"{name=a}] s0 s <-[gray, "$\beta $"{name=b}] r1 x; s0 x' <- {r0 x, r1 x}; }; \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (a) to (c); \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (a) to (d); \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (b) to (c); \draw [/tikz/commutative diagrams/Rightarrow, shorten >=-4pt] (b) to (d); \end {tikzpicture} ]]> while its weak collapse, whose index has two objects and four morphisms, essentially additionally identifying all the indices of above, is as follows: ] (x) .. controls (0,1,-1) .. node[above] {$\beta $} (s); \draw [gray, ->] (x) .. controls (0,-1,-1) .. node[below] {$\alpha $} (s); \draw [->] (x) .. controls (1,0,-1) .. node[right] {$\delta $.} (s); \draw [->] (x) .. controls (-1,0,-1) .. node[left] {$\gamma $} (s); \draw [every edge/.append style={shorten >=5pt, shorten <=5pt}] (0,0.8,-1) edge[commutative diagrams/Rightarrow, bend right] (-0.8,0,-1) (0,0.8,-1) edge[commutative diagrams/Rightarrow, bend left] (0.8,0,-1) (0,-0.8,-1) edge[commutative diagrams/Rightarrow, bend left] (-0.8,0,-1) (0,-0.8,-1) edge[commutative diagrams/Rightarrow, bend right] (0.8,0,-1) ; \end {tikzpicture} ]]> Note that the weak collapse is no longer simple, and we have drawn it in a non-planar fashion. The string diagram to which this is associated to is the following, which illustrates a scalar (see example ) in horizontal composition with a weak identity: Well-formed tree-cover oplax diagrams admit a special kind of colimit, using the strongly collapsible morphisms to induce a marking. Nick Hu20241121https://forest.nickx.hu/coherent-inverses-ZPGB/coherent-inverses-ZPGB/coherent-inverses-ZPGB/collapse colimitdefinition Let be a well-formed tree-cover oplax diagram. Define the marking as the closure under composition of the class of morphisms of which are strongly collapsible. We call the marked colimit the collapse colimit of . Diagrams in framed zigzag enriched categories when exploded naturally inherit a tree cover, which we explore in the following section. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-explosion/coherent-inverses-explosion/coherent-inverses-explosion/Exploded diagrams of framed zigzags and the level-wise tree-cover oplax diagramsection First, we introduce some machinery to allow us to describe a framed zigzag as data that is stratified into multiple dimensions by explosion, considering their representation as simple oplax diagrams. This stems fundamentally from the defining adjunction of the framed zigzag functor. Nick Hu2024103https://forest.nickx.hu/coherent-inverses-4S4Y/coherent-inverses-4S4Y/coherent-inverses-4S4Y/exploded diagramdefinition By lemma , we have a bijection between -functors and morphisms . With the adjunction of theorem , we obtain a composite bijection: For , define so that for all we have a -functor These data constitute the -fold exploded diagram of , with being the fully exploded diagram. Nick Hu20241212https://forest.nickx.hu/coherent-inverses-G1R1/coherent-inverses-G1R1/coherent-inverses-G1R1/explosion of simple oplax diagram is a simple oplax diagramlemma The explosion of a simple oplax diagram is a simple oplax diagram. Nick Hu20241229https://forest.nickx.hu/coherent-inverses-DZP6/coherent-inverses-DZP6/coherent-inverses-DZP6/exploded diagrams of framed zigzags and framed zigzag maps are simple oplax diagramscorollary Consider some framed zigzag . As an object of a -category, can be identified with a global element of , , a simple oplax diagram. The (fully) exploded diagram associated to is moreover a simple oplax diagram. The same holds for framed zigzag maps, treating in as . Nick Hu2024103https://forest.nickx.hu/coherent-inverses-OC2T/coherent-inverses-OC2T/coherent-inverses-OC2T/explosion bundledefinition For , recall that the objects of the -category are, for each , of the form and for and where is a framed zigzag of length . The mapping , extends to a ‘parenting’ -functor For , define in this manner. Thus, we have a sequence of ‘parenting’ -functors: which we call the explosion bundle of . Nick Hu20241014https://forest.nickx.hu/coherent-inverses-8DI9/coherent-inverses-8DI9/coherent-inverses-8DI9/explosion and parentingexample Let be a framed zigzag of singular height 1, which corresponds to the cospan in . Moreover, let and have singular height 0, corresponding to the 0-length iterated cospan given by the object , and have singular height 1 corresponding to the cospan . As an explosion bundle, we can illustrate (eliding most 1-cell labels and 2-cell fillers for clarity) the situation as the following: s <- x }; { [name=r0, fresh nodes] "" -!- x }; r1 x -> s0 s <- r0 x; r1 x -> s0 x <- r0 x; r1 x -> s0 x' <- r0 x; }; \end {tikzpicture} & \begin {tikzpicture}[baseline=(current bounding box.center), every node/.append style={draw=none}] \node [draw=gray] [yshift=0.5cm] {$\operatorname {Zig}(\mathcal {C})$}; \path graph[math nodes] { T; C; S; T ->["$b$"] C <-["$f$"] S; }; \end {tikzpicture} \ar [l, rightsquigarrow, "\text {explode}"'] & \begin {tikzpicture} \node [draw=none] (Z) {$Z$}; \node [above=0.25cm of Z] (c) {$\operatorname {Zig}^{2}(\mathcal {C})$}; \end {tikzpicture} \ar [l, rightsquigarrow, "\text {explode}"'] \end {tikzcd} ]]> for diagrams in , , and , respectively, indexed by categories s / "(s_0, s_0, r_0)" <- x / "(r_1, s_0, r_0)" }; { [name=r0, fresh nodes] "\phantom {(r_0, s_0, r_0)}" -!- x / "(r_0, r_0, r_0)" }; r1 x -> s0 s <- r0 x; r1 x -> s0 x <- r0 x; r1 x -> s0 x' <- r0 x; }; \end {tikzpicture} \ar [r, shorten = 5pt, "P_{Z}^2"] & \begin {tikzpicture}[baseline=(current bounding box.center), every node/.append style={draw=none}] \node [draw=gray] [yshift=0.5cm] {$E_{Z} (\mathbf {1})$}; \path graph[math nodes] { T / "(r_1, r_0)"; C / "(s_0, r_0)"; S / "(r_0, r_0)"; T -> C <- S; }; \end {tikzpicture} \ar [r, shorten = 5pt, "P_{Z}"] & \begin {tikzpicture} \node [draw=none] (Z) {$r_0$}; \node [above=0.25cm of Z] (c) {$\mathbf {1}$}; \end {tikzpicture} \end {tikzcd} ]]> where the parenting -functor is the (non-injective) tuple-tail map on objects, i.e. . This stratification forms a tree-cover oplax diagram, which we call the level-wise tree-cover oplax diagram. Nick Hu20241212https://forest.nickx.hu/coherent-inverses-LQGR/coherent-inverses-LQGR/coherent-inverses-LQGR/level-wise tree-cover oplax diagramsdefinition Let be a tree-cover oplax diagram. Then we define the level-wise tree-cover oplax diagram given by explosion of : where is the tree-cover obtained by extending along . That is, for each , we have a corresponding set given by preimage, and also an extra level to ensure that each for . The Hasse diagram under the inclusion ordering of looks like a tree with additional depth one compared to . The -fold level-wise tree-cover oplax diagram of a framed zigzag is of particular importance, whereby we start with a framed zigzag and consider it as a global element, as in corollary , equipping it with the trivial tree cover (remark ), and through explosion obtain some simple tree-cover oplax diagram where solely encodes the framed zigzag structure. Nick Hu20241231https://forest.nickx.hu/coherent-inverses-7DES/coherent-inverses-7DES/coherent-inverses-7DES/level-wise tree-cover oplax diagram of fully exploded diagramsexample Both the tree-cover oplax diagrams in example and example are simple level-wise tree-cover oplax diagrams obtained from the 2-fold explosion of some (distinct) , whose 1-fold explosion corresponds to some simple oplax diagram equipped with the trivial tree cover. Nick Hu20241231https://forest.nickx.hu/coherent-inverses-VEEU/coherent-inverses-VEEU/coherent-inverses-VEEU/level-wise tree-cover oplax diagram for the 3D monoid multiplication associatorexample The following illustrates the level-wise tree-cover oplax diagram obtained by the fully exploded diagram of the simple oplax diagram associated to the object of , for some appropriate (see section ), which encodes the monoid multiplication associator: f[x=3] <- x[x=4]}; {[name=s1, fresh nodes, y=3] x[x=2] -> m[x=3] <- x[x=4]}; {[name=r1, fresh nodes, y=2] x[x=1] -> f[x=2] <- x[x=3] -> f[x=4] <- x[x=5]}; {[name=s0, fresh nodes, y=1] x[x=1] -> m[x=2] <- x[x=3] -> f[x=4] <- x[x=5]}; {[name=r0, fresh nodes, y=0] x[x=0] -> f[x=1] <- x[x=2] -> f[x=3] <- x[x=4] -> f[x=5] <- x[x=6]}; s0 m <- {r0 f, r0 f', r1 f}; s0 m <-[gray] {r0 x, r0 x', r0 x'', r1 x, r1 x'}; s0 f <- {r0 f'', r1 f'}; s0 f <-[gray] {r0 x'', r0 x''', r1 x', r1 x''}; s1 m <- {r1 f, r1 f', r2 f}; s1 m <-[gray] {r1 x, r1 x', r1 x'', r2 x, r2 x'}; s0 x <- {r0 x, r1 x}; s0 x' <- {r0 x'', r1 x'}; s0 x'' <- {r0 x''', r1 x''}; s1 x <- {r1 x, r2 x}; s1 x' <- {r1 x'', r2 x'}; }; \end {scope} \node [draw, rounded corners, rotate fit=45, fit={(r0 x) (r0 x''')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(s0 x) (s0 x'')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r1 x) (r1 x'')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(s1 x) (s1 x')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r2 x) (r2 x')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r0 x) (r0 x''') (r2 x) (r2 x')}, inner sep=4pt] {}; \coordinate (b) at (r1 x'); \begin {scope}[canvas is zx plane at y=4, xscale=2] \path graph[math nodes, no placement, edge quotes={fill=white, inner sep=1pt, anchor=center, font=\tiny }] { {[name=r1, fresh nodes, y=2] x[x=2] -> f[x=3] <- x[x=4]}; {[name=s0, fresh nodes, y=1] x[x=2] -> a / "\alpha " [x=3] <- x[x=4]}; {[name=r0, fresh nodes, y=0] x[x=0] -> f[x=1] <- x[x=2] -> f[x=3] <- x[x=4] -> f[x=5] <- x[x=6]}; s0 a <- {r0 f, r0 f', r0 f'', r1 f}; s0 a <-[gray] {r0 x, r0 x', r0 x'', r0 x''', r1 x, r1 x'}; s0 x <- {r0 x, r1 x}; s0 x' <- {r0 x''', r1 x'}; }; \end {scope} \node [draw, rounded corners, rotate fit=45, fit={(r0 x) (r0 x''')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(s0 x) (s0 x')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r1 x) (r1 x')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r0 x) (r0 x''') (r1 x) (r1 x')}, inner sep=4pt] {}; \coordinate (a) at (s0 a); \begin {scope}[canvas is zx plane at y=10, xscale=2, xshift=3cm] \path graph[math nodes, no placement, edge quotes={fill=white, inner sep=1pt, anchor=center, font=\tiny }] { {[name=r2, fresh nodes, y=4] x[x=2] -> f[x=3] <- x[x=4]}; {[name=s1, fresh nodes, y=3] x[x=2] -> m[x=3] <- x[x=4]}; {[name=r1, fresh nodes, y=2] x[x=1] -> f[x=2] <- x[x=3] -> f[x=4] <- x[x=5]}; {[name=s0, fresh nodes, y=1] x[x=1] -> f[x=2] <- x[x=3] -> m[x=4] <- x[x=5]}; {[name=r0, fresh nodes, y=0] x[x=0] -> f[x=1] <- x[x=2] -> f[x=3] <- x[x=4] -> f[x=5] <- x[x=6]}; s0 f <- {r0 f, r1 f}; s0 f <-[gray] {r0 x, r0 x', r1 x, r1 x'}; s0 m <- {r0 f', r0 f'', r1 f'}; s0 m <-[gray] {r0 x', r0 x'', r0 x''', r1 x', r1 x''}; s1 m <- {r1 f, r1 f', r2 f}; s1 m <-[gray] {r1 x, r1 x', r1 x'', r2 x, r2 x'}; s0 x <- {r0 x, r1 x}; s0 x' <- {r0 x', r1 x'}; s0 x'' <- {r0 x''', r1 x''}; s1 x <- {r1 x, r2 x}; s1 x' <- {r1 x'', r2 x'}; }; \end {scope} \node [draw, rounded corners, rotate fit=45, fit={(r0 x) (r0 x''')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(s0 x) (s0 x'')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r1 x) (r1 x'')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(s1 x) (s1 x')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r2 x) (r2 x')}, inner sep=0pt] {}; \node [draw, rounded corners, rotate fit=45, fit={(r0 x) (r0 x''') (r2 x) (r2 x')}, inner sep=4pt] {}; \coordinate (t) at (r1 x'); \draw [->, shorten >=35pt, shorten <=60pt] (t) -> node[right, pos=0.6, rotate=45] {\ldots } (a); \draw [->, shorten >=35pt, shorten <=60pt] (b) -> node[right, pos=0.6, rotate=45] {\ldots } (a); \end {tikzpicture} ]]> Many arrows aligned along the Z-axis have been omitted for clarity, as well as 2-cell fillers. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-signature/coherent-inverses-signature/coherent-inverses-signature/framed zigzags over a signaturesection In this section, we expand upon the idea that the -fold iterated framed zigzag enriched category is like a ‘space’ of combinatorial encodings of -dimensional string diagrams with respect to some algebraic signature presented by , a graded -category. To highlight notationally the change of setting from the abstract to the concrete, we use in place of . Some prerequisite notions are first required for an appropriate algebraic setting, which we first recall for ordinary zigzags. Previous iterations of the theory of zigzags (without framing) only considered the case where was a poset: either of generators under boundary inclusion, or even more trivially as for being the maximum dimension of a generator in . Nick Hu2024102https://forest.nickx.hu/coherent-inverses-G6FI/coherent-inverses-G6FI/coherent-inverses-G6FI/source and target of a zigzagdefinition Every zigzag corresponds to some iterated cospan in , where has singular height . The source of is the object of , which we denote by , and the target is the object , denoted by . In the case where , i.e. there are no singular levels , the source and target are identical and represents the identity on . Two zigzags have the same boundary when their source and target match. Nick Hu2024101https://forest.nickx.hu/coherent-inverses-LFED/coherent-inverses-LFED/coherent-inverses-LFED/globular zigzagdefinition A zigzag in an iterated zigzag category is globular with respect to if or otherwise if every zigzag in the exploded diagram in associated to is globular with respect to ; every zigzag map in the 2-fold exploded diagram in between regular levels is an identity morphism. Globularity imposes the following condition: for any zigzag of dimension at least two, the source of the source of is equal to the source of the target, and the target of the target is equal to the target of the source. This is crucial to the interpretation of zigzags as combinatorial encodings of -dimensional string diagrams, i.e. representing terms of a globular -category. These notions generalise easily to the enriched setting, considering instead framed zigzag enriched categories. For the remainder of this chapter, we will consider all framed zigzags to be globular. Nick Hu20241017https://forest.nickx.hu/coherent-inverses-G4B5/coherent-inverses-G4B5/coherent-inverses-G4B5/oplax filtereddefinition A -category is oplax filtered if every finite diagram in has an oplax cocone. A -functor is oplax filtered if its domain is oplax filtered. Nick Hu2024101https://forest.nickx.hu/coherent-inverses-US67/coherent-inverses-US67/coherent-inverses-US67/signature and framingdefinition A signature is a graded -category whose objects are called generators, where the rank of a generator is called its dimension, and whose morphisms are called frames. Every generator is associated to a globular framed zigzag called the diagram of , satisfying the following. . 0]]> Treating as a global element , the exploded diagram is of the form viz. , and . Every diagram in the explosion bundle associated to is oplax filtered, with the fully exploded diagram admitting as the tip of an oplax cocone in . This oplax cocone is called the framing of , consisting of a collection of frames. A generator is well-formed if every diagram in the explosion bundle associated to is well-formed under the level-wise tree-cover, and the signature is well-formed when all of its generators are. Each generator of a signature is a generator in the algebraic sense: presents a globular -category, where is the maximal dimension of a generator in . The associated diagram is a combinatorial encoding of the term in that -category presented by , as an object of . Framing captures the idea that not only does every generator need to be tracked by its vertex weight in the -dimensional space given by the -graph underlying its fully exploded diagram, but moreover so does every incoming edge weight: how every other generator relates to (spacially). In particular, it determines the framed zigzag maps in definition ; while we don‘t specify this information in the definition of our recursive scheme, if we instead imagine an ’add new generator with specified source and target' operation, then these framed zigzag maps pin the freshly added new generator in the -dimensional space associated to the -graph underlying its fully exploded diagram, giving it a framing in the sense of topology. We give a few example signatures, showing how the framing is used to encode inverses of diagrams. Nick Hu202512https://forest.nickx.hu/coherent-inverses-9FVP/coherent-inverses-9FVP/coherent-inverses-9FVP/signature of an invertible 1-cellexample Let be the -category freely generated by models the algebraic signature which consists of a single 1-cell , whose diagram in , fully exploded, is given by the above as a cospan. The cospan represents its inverse: . Nick Hu202512https://forest.nickx.hu/coherent-inverses-IPBS/coherent-inverses-IPBS/coherent-inverses-IPBS/signature of a semigroupexample Let be the -category given by three objects , and non-trivial morphisms given by the Hom posets: models the algebraic signature which consists of a 2-cell semigroup with respect to a 1-cell endomorphism . When fully exploded, the diagram in is given similarly to example , and that of in is given by: ["$f_{r_0}$"] f <-["$f_{r_1}$"] x}; {[name=s0, fresh nodes, nodes={xshift=3cm}] x ->["$m_{(r_0, s_0)}$"] m <-["$m_{(r_1, s_0)}$"] x}; {[name=r0, fresh nodes] x ->["$f_{r_0}$"] f <-["$f_{r_1}$"] x ->["$f_{r_0}$"] f <-["$f_{r_1}$"] x / "x."}; r0 x -> s0 x <- r1 x; r0 x'' -> s0 x' <- r1 x'; r0 f ->["$m_{(s_0, r_0)}$"] s0 m; r0 f' ->["$m_{(s_1, r_0)}$"] s0 m; r1 f ->["$m_{(s_0, r_1)}$"] s0 m; {[edges={gray}] r0 x ->["$m_{(r_0, r_0)}$"{name=BL}] s0 m; r0 x' ->["$m_{(r_1, r_0)}$"{name=BM}] s0 m; r0 x'' ->["$m_{(r_2, r_0)}$"{name=BR}] s0 m; r1 x ->["$m_{(r_0, r_1)}$"{name=TL}] s0 m; r1 x' ->["$m_{(r_1, r_1)}$"{name=TR}] s0 m; }; }; \draw [every edge/.append style={commutative diagrams/Rightarrow, shorten <=3pt, shorten >=3pt}] (BL) edge (s0 x) (BL) edge (r0 f) (BM) edge (r0 f) (BM) edge (r0 f') (BR) edge (r0 f') (BR) edge (s0 x') (TL) edge (s0 x) (TL) edge (r1 f) (TR) edge (r1 f) (TR) edge (s0 x') ; \end {tikzpicture} ]]>Nick Hu202512https://forest.nickx.hu/coherent-inverses-RKCS/coherent-inverses-RKCS/coherent-inverses-RKCS/signature of a scalarexample Let be the -category given by two objects, , with non-trivial morphisms given by the Hom poset: models the algebraic signature which consists of a 2-cell scalar with respect to a 0-cell . When fully exploded, the diagram in is given by: ["$s_{(r_0, s_0)}$"{name=L}] s <-["$s_{(r_1, s_0)}$"{name=R}] x}; {[name=r0, fresh nodes, nodes={xshift=1.75cm}] x / "x."}; r0 x -> s0 x <- r1 x; r0 x -> s0 x' <- r1 x; {[edges={gray}] r0 x ->["$s_{(r_0, r_0)}$"{name=B}] s0 s; r1 x ->["$s_{(r_0, r_1)}$"{name=T}] s0 s; }; }; \draw [every edge/.append style={commutative diagrams/Rightarrow, shorten <=3pt, shorten >=3pt}] (B) edge (L) (B) edge (R) (T) edge (L) (T) edge (R) ; \end {tikzpicture} ]]> As a 2-cell, which can be composed both vertically and horizontally, comes in four variants, inverse in any combination with respect to this. The associated fully exploded diagrams in are represented using mirror symmetry along the respective axes: ["$s_{(r_0, s_0)}$"{name=L}] s <-["$s_{(r_1, s_0)}$"{name=R}] x}; {[name=r0, fresh nodes, nodes={xshift=1.75cm}] x}; r0 x -> s0 x <- r1 x; r0 x -> s0 x' <- r1 x; {[edges={gray}] r0 x ->["$s_{(r_0, r_0)}$"{name=B}] s0 s; r1 x ->["$s_{(r_0, r_1)}$"{name=T}] s0 s; }; }; \draw [every edge/.append style={commutative diagrams/Rightarrow, shorten <=3pt, shorten >=3pt}] (B) edge (L) (B) edge (R) (T) edge (L) (T) edge (R) ; \end {tikzpicture} \ar [r, leftrightsquigarrow, "\text {h. inv.}"] \ar [d, leftrightsquigarrow, "\text {v. inv.}"] & \begin {tikzpicture}[baseline=(current bounding box.center)] \path graph[math nodes, grow right=1.75cm, branch down=1.75cm, edge quotes={fill=white, inner sep=1pt, anchor=center, font=\tiny }] { {[name=r1, fresh nodes, nodes={xshift=1.75cm}] x}; {[name=s0, fresh nodes] x ->["$s_{(r_1, s_0)}$"{name=L}] s <-["$s_{(r_0, s_0)}$"{name=R}] x}; {[name=r0, fresh nodes, nodes={xshift=1.75cm}] x}; r0 x -> s0 x <- r1 x; r0 x -> s0 x' <- r1 x; {[edges={gray}] r0 x ->["$s_{(r_0, r_0)}$"{name=B}] s0 s; r1 x ->["$s_{(r_0, r_1)}$"{name=T}] s0 s; }; }; \draw [every edge/.append style={commutative diagrams/Rightarrow, shorten <=3pt, shorten >=3pt}] (B) edge (L) (B) edge (R) (T) edge (L) (T) edge (R) ; \end {tikzpicture} \ar [d, leftrightsquigarrow, "\text {v. inv.}"] \\ \begin {tikzpicture}[baseline=(current bounding box.center)] \path graph[math nodes, grow right=1.75cm, branch down=1.75cm, edge quotes={fill=white, inner sep=1pt, anchor=center, font=\tiny }] { {[name=r1, fresh nodes, nodes={xshift=1.75cm}] x}; {[name=s0, fresh nodes] x ->["$s_{(r_0, s_0)}$"{name=L}] s <-["$s_{(r_1, s_0)}$"{name=R}] x}; {[name=r0, fresh nodes, nodes={xshift=1.75cm}] x}; r0 x -> s0 x <- r1 x; r0 x -> s0 x' <- r1 x; {[edges={gray}] r0 x ->["$s_{(r_0, r_1)}$"{name=B}] s0 s; r1 x ->["$s_{(r_0, r_0)}$"{name=T}] s0 s; }; }; \draw [every edge/.append style={commutative diagrams/Rightarrow, shorten <=3pt, shorten >=3pt}] (B) edge (L) (B) edge (R) (T) edge (L) (T) edge (R) ; \end {tikzpicture} \ar [r, leftrightsquigarrow, "\text {h. inv.}"] & \begin {tikzpicture}[baseline=(current bounding box.center)] \path graph[math nodes, grow right=1.75cm, branch down=1.75cm, edge quotes={fill=white, inner sep=1pt, anchor=center, font=\tiny }] { {[name=r1, fresh nodes, nodes={xshift=1.75cm}] x}; {[name=s0, fresh nodes] x ->["$s_{(r_1, s_0)}$"{name=L}] s <-["$s_{(r_0, s_0)}$"{name=R}] x}; {[name=r0, fresh nodes, nodes={xshift=1.75cm}] x / "x."}; r0 x -> s0 x <- r1 x; r0 x -> s0 x' <- r1 x; {[edges={gray}] r0 x ->["$s_{(r_0, r_1)}$"{name=B}] s0 s; r1 x ->["$s_{(r_0, r_0)}$"{name=T}] s0 s; }; }; \draw [every edge/.append style={commutative diagrams/Rightarrow, shorten <=3pt, shorten >=3pt}] (B) edge (L) (B) edge (R) (T) edge (L) (T) edge (R) ; \end {tikzpicture} \end {tikzcd} ]]> We can relate to the previous iteration of the theory of zigzags as follows. Nick Hu20241231https://forest.nickx.hu/coherent-inverses-01J5/coherent-inverses-01J5/coherent-inverses-01J5/Unframingremark Using definition , for any signature , its underlying poset as a category forms the ‘unframed’ version of , as in the original zigzag category construction. Here, for any pair of generators, exactly if appears in some boundary of , although the framing — the information determining its position on the boundary — has been lost. Fundamentally, this is why the original zigzag category construction cannot capture the distinction between some combinatorial encoding of some -dimensional string diagram and its inverse, because this positional information is crucial. We also have a notion of minimal subsignature with respect to some set of generators. Nick Hu2024116https://forest.nickx.hu/coherent-inverses-LRMM/coherent-inverses-LRMM/coherent-inverses-LRMM/subsignaturedefinition Let be a signature, and be a subset of objects of . Then we call the full sub--category whose objects are given by the downwards closure of with respect to the subsignature of with respect to . There is a natural corestriction operator to produce a directly surjective -functor over its minimal subsignature from another -functor over a signature by obtaining from its image. Nick Hu2024116https://forest.nickx.hu/coherent-inverses-1I0W/coherent-inverses-1I0W/coherent-inverses-1I0W/Minimal subsignature corestriction of a -functorlemma Given any -functor , where is a graded -category, factors as =10pt, ""'] \end {tikzcd} ]]> where is directly surjective, and is the set of objects in the image of . Similarly, we can do the same with -functors over framed zigzag enriched categories. Nick Hu2024116https://forest.nickx.hu/coherent-inverses-SSX4/coherent-inverses-SSX4/coherent-inverses-SSX4/Minimal subsignature corestriction of a -functor over framed zigzagslemma Every diagram , where is a graded -category, factors as =10pt, ""'] \end {tikzcd} ]]> where is directly surjective, and is the set of objects in the image of . We use this to avoid having to talk about diagrams-with-respect-to-a-signature (à la terms-in-context). Operationally, this reflects the fact that many of our algorithms are not parametrised by signature. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-algorithms/coherent-inverses-algorithms/coherent-inverses-algorithms/Algorithmssection Here, we describe the core algorithms underlying homotopy.io and its manipulation of fully exploded diagrams. It suffices to represent such as -graphs as opposed to oplax diagrams, because none of our procedures require composition of morphisms; 2-cell filler data is property, not structure: it is either present or not, so it does not require explicit representation. Recall from the definition of oplax diagram that 2-cell fillers exist when a path may be ‘shortcut’ into another path, in the direction of shorter path to longer path. Another change in representation is for tree covers, which are instead represented by a depth function, assigning to an edge the rank of its smallest containing set with respect to the inclusion ordering. This is strictly less information, but suffices for our purposes. We also slowly begin to introduce -dimensional string diagrammatic aspects in place of, or in addition to, drawing fully exploded diagrams, as in figure . Nick Hu202512https://forest.nickx.hu/coherent-inverses-J4C6/coherent-inverses-J4C6/coherent-inverses-J4C6/2D string diagram versus fully exploded diagram in figure] (xb) edge (xl) (xt) edge (xl) (xb) edge (xr) (xt) edge (xr) ; \draw[->] (xl) edge["$s_{(r_0, s_0)}$"{commutative diagrams/description, name=L}] (s) (xr) edge["$s_{(r_1, s_0)}$"{commutative diagrams/description, name=R}] (s) (xb) edge["$s_{(r_0, r_0)}$"{commutative diagrams/description, name=B}] (s) (xt) edge["$s_{(r_0, r_1)}$"{commutative diagrams/description, name=T}] (s) ; \draw[every edge/.append style={commutative diagrams/Rightarrow, shorten <=3pt, shorten >=3pt}] (B) edge (L) (B) edge (R) (T) edge (L) (T) edge (R) ; \end{tikzpicture} ]]> Fully exploded diagram for scalar overlaid on its 2D string diagram. The first basic procedure for explosion is as follows. Nick Hu20241128https://forest.nickx.hu/coherent-inverses-BAGY/coherent-inverses-BAGY/coherent-inverses-BAGY/explosionalgorithm 0$, depth function $d\colon E \to \mathbb {N}$ \Ensure weighted directed graph $G^\prime = (V^\prime , E^\prime , w^\prime )$ of dimension $n - 1$, depth function $d^\prime \colon E^\prime \to \mathbb {N}$ \Function {Explode}{$G$, $d$} \State $G^\prime \gets $ empty graph \State $d^\prime \gets $ empty function \For {$v \in V$} \Comment {explode vertex $v$} \LComment {$w(v)$ is a weighted directed graph of dimension $n - 1$} \State $G_v = (V_v, E_v, w_v) \gets w(v)$ \For {$v_v \in V_v$} \State \Call {AddNode}{$G^\prime $, $w_v(v_v)$} \EndFor \For {$e_v = (u_v, v_v) \in E_v$} \State \Call {AddEdge}{$G^\prime $, $u_v$, $v_v$, $w_v(e_v)$} \State $d^\prime (e_v) \gets d((v_v, v_v))$ \EndFor \EndFor \For {$e = (u, v) \in E$} \Comment {explode edge $e$} \State $G_u = (V_u, E_u, w_u) \gets w(u)$ \State $G_v = (V_v, E_v, w_v) \gets w(v)$ \LComment {$w(e)$ is a weighted directed graph homomorphism of dimension $n - 1$, i.e. a map $V_u \to V_v$ equipped with a weighting $w_e$} \State $\left (h\colon G_u \to G_v, w_e\right ) \gets w(e)$ \For {$v_u \mapsto v_v \in h$} \State \Call {AddEdge}{$G^\prime $, $v_u$, $v_v$, $w_e((v_u, v_v))$} \State $d^\prime ((v_u, v_v)) \gets d(e) + 1$ \EndFor \EndFor \Return $(G^\prime , d^\prime )$ \EndFunction \end {algorithmic}} ]]> This constructs an exploded diagram along with its level-wise tree-cover, and can be iterated to obtain a fully exploded diagram. As an algorithm, we express weak collapse as in algorithm . Nick Hu20241127https://forest.nickx.hu/coherent-inverses-FEO0/coherent-inverses-FEO0/coherent-inverses-FEO0/collapsealgorithm Note that we have not used any data associated to the tree cover, i.e. the depth function. In contrast, the variant for strong collapse is as in algorithm . Nick Hu20241128https://forest.nickx.hu/coherent-inverses-VQWY/coherent-inverses-VQWY/coherent-inverses-VQWY/depth-first collapsealgorithm An example that highlights the differences is given in example . In the following example, both coincide. Nick Hu2024122https://forest.nickx.hu/coherent-inverses-W55X/coherent-inverses-W55X/coherent-inverses-W55X/Snake collapseexample Fix a signature comprising two 0-cells and a single invertible 1-cell . In this signature, recall that we have cancellation and introduction 2-cells: From this, we can build a 'snake' composite as the following: where denotes vertical composition of 2-cells. As a 2D string diagram, such a composite looks like the following: where and are represented by blue and red regions respectively, and is the yellow wire connecting them ( is indicated by a different shade of yellow, with cancellations and introductions given by yellow dots). This diagram, as an object of , has combinatorial structure given by the following -graph (via explosion): ["$f_{r_0}$"] f <-["$f_{r_1}$"] y}; {[name=s1, fresh nodes, nodes={xshift=1.5cm}] x ->["$f_{r_0}$"] f <-["$f_{r_0}$"] x [xshift=1.5cm] ->["$f_{r_0}$"] f [xshift=1.5cm] <-["$f_{r_1}$"] y [xshift=1.5cm]}; {[name=r1, fresh nodes] x ->["$f_{r_0}$"] f <-["$f_{r_1}$"] y ->["$f_{r_1}$"] f <-["$f_{r_0}$"] x ->["$f_{r_0}$"] f <-["$f_{r_1}$"] y}; {[name=s0, fresh nodes] x ->["$f_{r_0}$"] f <-["$f_{r_1}$"] y ->["$f_{r_1}$"] f [xshift=1.5cm] <-["$f_{r_1}$"] y [xshift=1.5cm]}; {[name=r0, fresh nodes] x ->["$f_{r_0}$"] f <-["$f_{r_1}$"] y}; {[edges={orange, postaction={decorate}}] r0 f -> s0 f <- r1 f -> s1 f <- r1 f' -> s0 f' <- r1 f'' -> s1 f' <- r2 f}; {[edges={cyan, postaction={decorate}}] r0 x -> s0 x <- r1 x -> s1 x <- r2 x}; {[edges={red, postaction={decorate}}] r0 y -> s0 y' <- r1 y' -> s1 y <- r2 y}; {[edges={red, postaction={decorate}}] r0 y -> s0 y <- r1 y}; {[edges={cyan, postaction={decorate}}] r1 x' -> s1 x' <- r2 x}; {[edges={gray, "$f_{r_0}$"}] r0 x -> s0 f <- r1 x -> s1 f <- r2 x -> s1 f' <- r1 x' -> {s0 f', s1 f}}; {[edges={gray, "$f_{r_1}$"}] r2 y -> s1 f' <- r1 y' -> s0 f' <- r0 y -> s0 f <- r1 y -> {s1 f, s0 f'}}; }; \end {tikzpicture} ]]> Every atomic edge is coloured (recall that an atomic edge is one which appears as an outer face, i.e. it is not a 'composite'), and non-atomic edges are grayed out. Each triangle, as in algorithm line 6, is just a collection of triples of edges in this -graph which form a triangle. It happens to be the case that every such triangle admits the identity as its underlying 2-cell filler, thus every atomic identity edge is collapsible, which we mark by crossing out. The resulting collapsed -graph is given by: The collapse algorithm is used in the construction of signatures in order to ensure well-formedness, as follows. Nick Hu20241128https://forest.nickx.hu/coherent-inverses-9DJY/coherent-inverses-9DJY/coherent-inverses-9DJY/generator creationalgorithm 0$ \Ensure weighted directed graph $G = (V, E, w)$ \Function {GenerateBoundary}{$s, t$} \State $\mathsf {src} \gets s$; $\mathsf {tgt} \gets t$ \State $G \gets \set {\mathsf {src}} \cup \set {\mathsf {tgt}}$ \While {$\Call {Dim}{G} > 0$} \State $(G, \_) \gets \Call {Explode}{G, \emptyset }$ \For {$\partial \in \set {\Call {Source}{}, \Call {Target}{}}$} \State $v \gets \Call {AddNode}{G, \Call {\partial }{\mathsf {src}}}$ \Comment {globularity: $\Call {\partial }{\mathsf {src}} = \Call {\partial }{\mathsf {tgt}}$} \State $\Call {AddEdge}{G, \Call {\partial }{\mathsf {src}}, v, \text {id}_{} }$ \State $\Call {AddEdge}{G, \Call {\partial }{\mathsf {tgt}}, v, \text {id}_{} }$ \EndFor \State $\mathsf {src} \gets \Call {Source}{\mathsf {src}}$; $\mathsf {tgt} \gets \Call {Target}{\mathsf {tgt}}$ \EndWhile \Return $G$ \EndFunction \Require signature $\Sigma $, new generator name $\sigma $, source $s$, target $t$ \Statex where $n \coloneqq \Call {Rank}{s} = \Call {Rank}{t}$ \Statex satisfying globularity for $s$ and $t$ if $n > 0$ \Ensure extended signature $\Sigma \subset \Sigma ^\prime $ \Function {AddGeneratorN}{$\Sigma $, $\sigma $, $s$, $t$} \State $D \gets \Call {GenerateBoundary}{s, t}$ \LComment {will make an $(n+1)$-dimensional diagram $D$ for $\sigma $ which looks like an $(n+1)$-cube with $s$ and $t$ as $n$-dimensional opposing faces} \State $ns \gets \Call {Nodes}{D}$ \State \Call {AddNode}{$D$, $\sigma $} \LComment {generate frame identifications} \State {$G_s \gets \set {s}$; $G_t \gets \set {t}$} \While {$\Call {Dim}{G_s} > 0$} \State $(G_s, \_) \gets \Call {Explode}{G_s, \emptyset }$; $(G_t, \_) \gets \Call {Explode}{G_t, \emptyset }$ \EndWhile \State $(\_, \equiv _s) \gets \Call {Collapse}{G_s}$; $(\_, \equiv _t) \gets \Call {Collapse}{G_t}$ \State $\equiv \gets $ trivial equivalence relation on $ns$ \State $\equiv \gets \equiv \cup \equiv _s \cup \equiv _t$ \State $M \gets $ map keyed by equivalence classes of $\equiv $, values all distinct \For {$n \in ns$} \Comment {add frames} \State \Call {AddEdge}{$D$, $n$, $\sigma $, \Call {Get}{$M$, ${[n]}_\equiv $}} \EndFor \LComment {weight of edge $u \to \sigma $ is the same as that of $v \to \sigma $ exactly when $u$ and $v$ appear on the same source/target boundary and are identified by \Call {Collapse}{} on that boundary} \State \Return $\Sigma \cup \set {\sigma \mapsto D}$ \EndFunction \end {algorithmic}} ]]> Note that lines 18–28 determine frame identifications for the new generator in a non-trivial way, so as to ensure that the result is well-formed. Crucially, we use the version of the collapse algorithm which ignores depth, whose result is not necessarily a simple -graph, as example shows. Nick Hu20241128https://forest.nickx.hu/coherent-inverses-GV0W/coherent-inverses-GV0W/coherent-inverses-GV0W/Snake to identityexample Fix a signature as in example . We wish to demonstrate generator creation of a 3D generator whose source 2D diagram is the snake, and target 2D diagram is the identity on . As a 3D surface diagram, the diagram of such a generator (represented by the purple ball) is shown in figure . Nick Hu202519https://forest.nickx.hu/coherent-inverses-7M69/coherent-inverses-7M69/coherent-inverses-7M69/Algebraic snake-to-identity generator 3D surface diagramfigure A 3D surface diagram of the algebraic generator whose source (left) is the snake, and target (right) is the identity. This diagram is given by the framed zigzag , now ordering slices bottom-up, where each slice is a 2D string diagram: ["$a$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-2-1-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); {(5), (6)} ->["$b$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-2-1-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); {(7)} ->["$c$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-2-1-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); {(8)} ->["$d$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-2-1-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); }; \end {tikzpicture} ]]> Here, we have illustrated some of the framing data associated to . In particular, because the source diagram is the snake, which under the collapse algorithm determines 3 equivalence classes in the source boundary (which respectively associate to the blue region, the yellow wire, and the red region), each frame into originating from any point on the wire (which we have drawn with dashed arrows) is identified as . Similarly, the target diagram also identifies the two frames indicated as . The following example illustrates why weak collapse is the appropriate notion for the generator creation algorithm, as opposed to strong collapse. Nick Hu20241129https://forest.nickx.hu/coherent-inverses-FMG7/coherent-inverses-FMG7/coherent-inverses-FMG7/Necessity of weak collapse for generator creation algorithmexample Fix an algebraic signature consisting of the following generators: 0-cells , ; 1-cells ; 2-cells . Consider a 3D generator given as an endomorphism of the following 2D string diagram : Here, is represented by the purple dot, and the diagram as a whole represents the composite for being the vertical composition of 2-cells. As a -graph underlying a fully exploded diagram, looks like figure . No morphism in this diagram is collapsible. The 2-cell is represented by figure . is then given by the gluing of both along the common row. Nick Hu2025116https://forest.nickx.hu/coherent-inverses-ORQV/coherent-inverses-ORQV/coherent-inverses-ORQV/Fully exploded diagram of algebraic capfigure["$\gamma _{(r_0, s_0)}$"] c / "\gamma " <-["$\gamma _{(r_1, s_0)}$"] x }; {[name=r0, fresh nodes] x ->["$f_{(r_0)}$"] f <-["$f_{(r_1)}$"] x ->["$f_{(r_1)}$"] f <-["$f_{(r_0)}$"] x / "x." }; s0 x <- {r1 x, r0 x}; s0 x' <- {r1 x, r0 x''}; s0 c <- { r1 x [>"$\gamma _{(r_0, r_1)}$"], r0 x [>"$\gamma _{(r_0, r_0)}$"], r0 f [>"$\gamma _{(s_0, r_0)}$"], r0 x' [>"$\gamma _{(r_1, r_0)}$"], r0 f' [>"$\gamma _{(s_1, r_0)}$"], r0 x'' [>"$\gamma _{(r_2, r_0)}$"], }; }; \end {tikzpicture} ]]> as a fully exploded diagram in .Nick Hu2025116https://forest.nickx.hu/coherent-inverses-EJ8O/coherent-inverses-EJ8O/coherent-inverses-EJ8O/Fully exploded diagram of homotopy cupfigure["$f_{(r_0)}$"] f <-["$f_{(r_1)}$"] x ->["$f_{(r_1)}$"] f <-["$f_{(r_0)}$"] x }; {[name=s0, fresh nodes, nodes={xshift=2cm}] x ->["$f_{(r_0)}$"] c / "f" <-["$f_{(r_0)}$"] x }; {[name=r0, fresh nodes, nodes={xshift=4cm}] x }; s0 x <-[postaction={decorate}] {r0 x, r1 x}; s0 x' <-[postaction={decorate}] {r0 x, r1 x''}; s0 c <- { r0 x [>"$f_{(r_0)}$"], r1 x [>"$f_{(r_0)}$"], r1 f [> postaction={decorate}], r1 x' [>"$f_{(r_1)}$"], r1 f' [> postaction={decorate}], r1 x'' [>"$f_{(r_0)}$"], }; }; \end {tikzpicture} ]]> as a fully exploded diagram in , where crossed edges indicate strongly collapsible morphisms with respect to the level-wise tree-cover oplax diagram. Note that according to the bottom diagram (figure ), the two points labelled in the top diagram (figure ) should be identified, and similarly for the outermost s on the bottom row. This is the crux of why the generator creation algorithm is based on weak collapse rather than strong collapse. The resulting collapse causes the outermost s and s to be respectively identified on the bottom row, and the -graph underlying the weak collapse is then: ["$\gamma_{(r_0, s_0)}$"] c / "\gamma" <-["$\gamma_{(r_1, s_0)}$"] x }; {[name=r0, fresh nodes] x ->["$f_{(r_0)}$"] f <-["$f_{(r_1)}$"] x ->["$f_{(r_1)}$"] f <-["$f_{(r_0)}$"] x }; s0 x <- {r1 x, r0 x}; s0 x' <- {r1 x, r0 x''}; s0 c <- { r1 x [>"$\gamma_{(r_0, r_1)}$"], r0 x [>"$\gamma_{(r_0, r_0)}$"], r0 f [>"$\gamma_{(s_0, r_0)}$"], r0 x' [>"$\gamma_{(r_1, r_0)}$"], r0 f' [>"$\gamma_{(s_1, r_0)}$"], r0 x'' [>"$\gamma_{(r_2, r_0)}$"], }; r0 f <->[clear >, bend right, tips=false, double equal sign distance] r0 f'; r0 x <->[clear >, bend right, tips=false, double equal sign distance] r0 x''; }; }; \node[right=of left] (right) { \tikz \path graph[math nodes, grow right=2cm, edge quotes={fill=white, inner sep=1pt, anchor=center, font=\tiny}] { {[name=r2, fresh nodes] x}; {[name=s1, fresh nodes, nodes={xshift=-2cm}] x ->["$\gamma_{(r_0, s_0)}$"] c / "\gamma" <-["$\gamma_{(r_1, s_0)}$"] x}; {[name=r1, fresh nodes] x}; {[name=s0, fresh nodes] f}; {[name=r0, fresh nodes] x}; s1 x <- {r2 x, r1 x}; s1 x' <- {r2 x, r1 x}; s1 c <-["$\gamma_{(r_1, r_1)}$"] r2 x; s1 c <-["$\gamma_{(r_1, r_0)}$"] r1 x; s0 f ->[bend left, in=100, out=100, "$\gamma_{(s_0, r_0)}$"{pos=0.4}] s1 c; s0 f ->[bend right, in=-100, out=-100, "$\gamma_{(s_1, r_0)}$"{pos=0.4}] s1 c; r0 x ->[bend left, in=90, out=90, looseness=1.4, "$\gamma_{(r_0, r_0)}$"{near start}] s1 c; r0 x ->[bend right, in=-90, out=-90, looseness=1.4, "$\gamma_{(r_2, r_0)}$"{near start}] s1 c; s0 f <-["$f_{(r_1)}$"] r1 x; s0 f <-["$f_{(r_0)}$"] r0 x; }; }; \draw[->] (left) edge["redraw", commutative diagrams/rightsquigarrow] (right); \end {tikzpicture} ]]> Let denote the 3D generator which is an endomorphism on , pictorially represented by the green ball. As a 3D surface diagram, the diagram of such a generator is shown in figure . Nick Hu202519https://forest.nickx.hu/coherent-inverses-GZ9V/coherent-inverses-GZ9V/coherent-inverses-GZ9V/‘Hourglass’ generator 3D surface diagramfigure Endomorphism on . The following diagram illustrates the framing that this generator obtains: ["$a$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-2-1-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); {(4)} ->["$b$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-3-2-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); {(5), (6), (7)} ->["$c$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-2-1-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); {(8)} ->["$d$"{circle,fill=white,text=black,fill opacity=.5,text opacity=1,inner sep=1pt},line width=2pt,generator-3-2-0-pos,dashed,shorten <=6pt,shorten >=6pt] (s); }; \end {tikzpicture} ]]> Here, the frames and are the new morphisms and respectively (and similarly on the target boundary for and ) added to the signature by generator creation; that the three frames labelled are identified ensures well-formedness of this diagram in the sense that the three parallel paths are quotiented together. If instead we had used strong collapse on (which is the identity), focusing on the framing from the source of figure , we would obtain different frames , , and (which were all previously). This violates well-formedness of the resulting generator: restricting to its source boundary (which looks like and is a non-maximal tree of the level-wise tree-cover oplax diagram) and examining the portion which looks like figure , we see that any of the collapsible morphisms (when restricting to this tree, as in figure ) connecting a pair of s is no longer collapsible with respect to the whole diagram, precisely because . However, strong collapse is necessary for the contraction algorithm and the typechecking algorithm, which we now describe. Nick Hu20241128https://forest.nickx.hu/coherent-inverses-FHS4/coherent-inverses-FHS4/coherent-inverses-FHS4/contractionalgorithm 1$, $(i, i+1)$ such that $0 \leq i < h$ \Ensure contracting framed zigzag map $Z \xrightarrow {c} Z^\prime $ or failure \Function {ContractPrepare}{$Z$, ($i$, $i+1$)} \LComment {slice $Z$ to singular heights $i$ and $i+1$, and adjacent regular heights} \State $G \gets \set {r_i \xrightarrow {f_i} s_i \xleftarrow {b_i} r_{i+1} \xrightarrow {f_{i+1}} s_{i+1} \xleftarrow {b_{i+1}} r_{i+2}}$ as a weighted graph \State $d \gets $ constant function at $0$ \State $\gamma \gets $ \Call {Contract}{$G$, $d$} \Comment {$\gamma $ is a sub-framed zigzag map} \State \Return extension of $\gamma $ to $Z$, acting as $ \text {id}_{} $ everywhere else \EndFunction \Require weighted directed graph $G = (V, E, w)$, depth function $d\colon E \to \mathbb {N}$ \Ensure cocone over $G$ or failure \Function {Contract}{$G$, $d$} \If {\Call {Dim}{$G$} $= 0$} \State \Return \Call {ContractBase}{$G$, $d$} \ElsIf {\Call {Dim}{$G$} $> 0$} \State \Return \Call {ContractRecursive}{$G$, $d$} \EndIf \EndFunction \Require weighted directed graph $G = (V, E, w)$ of dimension $n = 0$, depth function $d\colon E \to \mathbb {N}$ \Ensure cocone over $G$ or failure \Function {ContractBase}{$G$, $d$} \State $(C = (V, E, w), \_) \gets \Call {DepthFirstCollapse}{G, d}$ \State $m \gets \max _{v \in V} \Call {Rank}{v}$ \State $\equiv \gets $ equivalence relation on $V$ where \Statex $\qquad u \equiv v \iff u = v \vee \Call {Rank}{u} = \Call {Rank}{v} = m$ \State $C / \equiv \gets \Call {Quotient}{C, \equiv }$ \If {$\exists $ terminal vertex $v$ of $C / \equiv $} \State \Return cocone over $G$ with tip $v$ \Else \State \textbf {fail} \EndIf \EndFunction \Require weighted directed graph $G = (V, E, w)$ of dimension $n > 0$, depth function $d\colon E \to \mathbb {N}$ \Ensure cocone over $G$ or failure \Function {ContractRecursive}{$G$, $d$} \State $cs \gets []$ \For {$G_k \in $ \Call {SingularDecomposition}{G}} \LComment {\textbf {fail} if \Call {SingularDecomposition}{G} fails} \State $(G^\prime _k, d^\prime _k) \gets $ \Call {Explode}{$G_k$, $d$} \State $c \gets $ \Call {Contract}{$G^\prime _k$, $d^\prime _k$} \Comment {\textbf {fail} if this fails} \State \Call {Append}{$cs$, $c$} \EndFor \LComment {each cocone tip $cs[i]$ corresponds to $r_i \xrightarrow {f_i} s_i \xleftarrow {b_i} r_{i+1}$ of the resulting contracted framed zigzag} \State $t \gets \Call {GlueTips}{cs}$ \Comment {glue these together to form the tip of a higher-dimensional cocone} \LComment {for a cocone $cs[i]$, its legs correspond to framed zigzag maps to $r_i, s_i, r_{i+1}$} \For{$i \in [0, \Call{Length}{cs} - 1)$} \For {$(f_0, g_0, h_0), (f_1, g_1, h_1) \gets \Call {Legs}{cs[i]}, \Call {Legs}{cs[i+1]}$} \State \Call {Assert}{$h_0 = f_1$} \Comment {check compatibility} \EndFor \EndFor \Return cocone over $G$ with tip $t$ \EndFunction \end {algorithmic}} ]]> This procedure is the primary component for the construction of complex zigzag homotopies, which provides an interface for a user of homotopy.io to build non-trivial proof terms (see section ). We argue for the correctness of this procedure in the following section. By basing the typechecking algorithm on the depth-first collapse algorithm, we ensure that higher-dimensional coherences of invertible generators also typecheck, as in example . Nick Hu20241128https://forest.nickx.hu/coherent-inverses-4D20/coherent-inverses-4D20/coherent-inverses-4D20/typecheckingalgorithm 0$} \Comment {fully explode $G$} \State $(G, d) \gets \Call {Explode}{G, d}$ \EndWhile \State $(C = (V, E, w), \_) \gets \Call {DepthFirstCollapse}{G, d}$ \If { \State $\forall k \in V. k$ is a sink of $C$. \Statex \textbf {let} $C_k$ be the closure of $k$ under reverse-reachability in $C$ \Statex $\exists \sigma \in \Sigma . \Call {IsGraphIso}{\Call {Diagram}{\sigma }, C_k}$ } \State \textbf {ok} \Else \State \textbf {fail} \EndIf \EndFunction \end {algorithmic}} ]]> This typechecking algorithm checks that the framed zigzag represents a valid term in the free globular -category presented by the algebraic signature . It does so by breaking up the fully exploded diagram of into pieces, and then ensuring that they match up with diagrams present in . Nick Hu2023123https://forest.nickx.hu/coherent-inverses-contraction-correctness/coherent-inverses-contraction-correctness/coherent-inverses-contraction-correctness/Correctness of framed contractionsection First, we define a subclass of framed zigzags: the zigzag homotopies. We demonstrate an argument for correctness of the contraction algorithm by showing that it always constructs zigzag homotopies (never failing) for certain classes of inputs. Nick Hu20241127https://forest.nickx.hu/coherent-inverses-OBFZ/coherent-inverses-OBFZ/coherent-inverses-OBFZ/zigzag homotopydefinition A zigzag homotopy between two framed zigzags , for 0]]>, is a cospan such that every 2-cell filler of the exploded diagrams of and in is trivial. These are a particular kind of framed zigzag in . A composite zigzag homotopy is an iterated cospan consisting of zigzag homotopies. The crux of this chapter is the following: the base case of the contraction algorithm computes a collapse colimit in the minimal subsignature of a well-formed level-wise tree-cover oplax diagram in some signature . Nick Hu202513https://forest.nickx.hu/coherent-inverses-063A/coherent-inverses-063A/coherent-inverses-063A/contraction algorithm base case computes a collapse colimittheorem Let be some well-formed tree-cover oplax diagram, where is a minimal subsignature, arising as the level-wise tree-cover oplax diagram obtained from the fully exploded diagram of some simple oplax diagram equipped with the trivial tree cover. admits a collapse colimit if and only if (algorithm ) succeeds, where encodes , encodes , and the result of the procedure encodes the collapse colimit. Nick Hu202513proof There is a bijective correspondence between marked cocones over (under a marking determined by strongly collapsible morphisms) and oplax cocones over the strong collapse given by duplicating/deleting legs. is a minimal subsignature, thus graded by definition. Hence an oplax cocone over exists exactly when there is a unique generator in the image of which is uniquely reachable (necessarily of maximal rank), which is precisely that which is constructed by . Nick Hu20241126https://forest.nickx.hu/coherent-inverses-M1IC/coherent-inverses-M1IC/coherent-inverses-M1IC/Juxtaposition with identity contractsproposition Let be a diagram of the following form: [tips=false, double equal sign distance] s1 / "r^\prime " <-[tips=false, double equal sign distance] r1 / "r^\prime " ->["$b$"] s0 / "s" <-["$f$"] r0 / "r."; }; \end {tikzpicture} ]]> Observe that this diagram corresponds the exploded diagram of a particular object of with two singular levels, where the second cospan is given by identity morphisms in . Then the contraction algorithm witnesses a framed zigzag map in whose exploded diagram is given as follows: [tips=false, double equal sign distance] s1 / "r^\prime " <-[tips=false, double equal sign distance] r1 / "r^\prime " ->["$b$"] s0 / "s" <-["$f$"] r0 / "r"}; { [name=r, nodes={yshift=-2cm}] r1 / "r^\prime " ->["$b$"] s0 / "s" <-["$f$"] r0 / "r,"}; l r2 ->[tips=false,double equal sign distance] r r1; {l r2 [< "$b$" right], l s1 [< "$b$" below], l r1 [< "$b$" below], l s0 [< {tips=false, double equal sign distance}], l r0 [< "$f$" above]} -> r s0; l r0 [clear <] ->[tips=false,double equal sign distance] r r0; }; \end {tikzpicture} ]]> where every 2-cell filler is trivial. Nick Hu20241126proof Suppose that , whence . Then the preimage of each identity morphism in is determined to be collapsible, and therefore the corresponding strong collapse is along the following quotient: [tips=false, double equal sign distance] s1 / "r^\prime " <-[tips=false, double equal sign distance] r1 / "r^\prime " ->["$b$"] s0 / "s" <-["$f$"] r0 / "r."; }; \node [draw, rounded corners, fit={(r2.center) (r1.center)}, inner sep=5pt] {}; \node [draw, rounded corners, fit={(s0.center)}, inner sep=5pt] {}; \node [draw, rounded corners, fit={(r0.center)}, inner sep=5pt] {}; \end {tikzpicture} ]]> The strong collapse admits a terminal index, which maps to ; the unique morphisms of to are respectively , , and , which assemble into a cocone with tip , lifting to form the desired framed zigzag map as an oplax cocone which witnesses a collapse colimit, as in theorem . Otherwise, when 0]]>, observe that this diagram admits a singular decomposition of into for having singular height . Each diagram yields an exploded diagram of the following form in : ["$h_0$"] 1 / "l^\prime _1" <-["$h_1$"] 2 / "{\ldots }" ->["$h_{n-2}$"] 3 / "l^\prime _{n-1}" <-["$h_{n-1}$"] 4 / "l^\prime _n"}; { [name=s1] 0 / "l^\prime _0" ->["$h_0$"] 1 / "l^\prime _1" <-["$h_1$"] 2 / "{\ldots }" ->["$h_{n-2}$"] 3 / "l^\prime _{n-1}" <-["$h_{n-1}$"] 4 / "l^\prime _n"}; { [name=r1] 0 / "l^\prime _0" ->["$h_0$"] 1 / "l^\prime _1" <-["$h_1$"] 2 / "{\ldots }" ->["$h_{n-2}$"] 3 / "l^\prime _{n-1}" <-["$h_{n-1}$"] 4 / "l^\prime _n"}; { [name=s0] 0 / "r_0" ->["$p$" above] 2 / "s_0" [xshift=2cm] <-["$q$" above] 4 / "r_1" [xshift=4cm]}; { [name=r0] 0 / "l_0" ->["$g_0$" below] 1 / "l_1" <-["$g_1$" below] 2 / "{\ldots }" ->["$g_{m-2}$" below] 3 / "l_{m-1}" <-["$g_{m-1}$" below] 4 / "l_m."}; \foreach \x [evaluate=\x as \xp using int(\x +1)] in {1} { \foreach \y in {1, 2, 3} { (s\x \space \y ) <-[tips=false, double equal sign distance] { (r\x \space \y ), (r\xp \space \y ) }; } }; (s1 1) <-["$h_0$" above] { r2 0, r1 0 }; (s1 1) <-["$h_1$" above] { r2 2, r1 2 }; (s1 3) <-["$h_{n-2}$" above] { r2 2, r1 2 }; (s1 3) <-["$h_{n-1}$" above] { r2 4, r1 4 }; \foreach \x [evaluate=\x as \xp using int(\x +1)] in {0, 1} { (s\x \space 0) <-[tips=false, double equal sign distance] { (r\x \space 0), (r\xp \space 0) }; (s\x \space 4) <-[tips=false, double equal sign distance] { (r\x \space 4), (r\xp \space 4) }; }; (s0 2) <-["$b_0$" below, ""{commutative diagrams/marking, name=a}] (r1 0); (s0 2) <-["$b_1$" right] (r1 1); (s0 2) <-[""{commutative diagrams/marking, name=b}] (r1 2); (s0 2) <-["$b_{n-1}$" right] (r1 3); (s0 2) <-["$b_n$" below, ""{commutative diagrams/marking, name=c}] (r1 4); (s0 2) <-["$f_0$" above, ""{commutative diagrams/marking, name=d}] (r0 0); (s0 2) <-["$f_1$" right] (r0 1); (s0 2) <-[""{commutative diagrams/marking, name=e}] (r0 2); (s0 2) <-["$f_{m-1}$" below] (r0 3); (s0 2) <-["$f_m$" above, ""{commutative diagrams/marking, name=f}] (r0 4); }; \begin {scope}[on background layer] \path (a) edge[commutative diagrams/Rightarrow] (s0 0); \path (a) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (s0 4); \path (d) edge[commutative diagrams/Rightarrow] (s0 0); \path (d) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (s0 4); \end {scope} \end {tikzpicture} ]]> Here, represents either some regular or singular level depending on whether is even or odd (the distinction here is unimportant). Now, we proceed inductively on the dimension . For the base case, where , this exploded diagram is a diagram in . Because was well-formed, so are each . Thus, the depth-first collapse algorithm quotient is first computed row-wise; from top to bottom: contiguous segments of are mapped to some for and for some : as the first three rows are identical, by well-formedness each computation in the three rows is identical — a priori, this may not be so due possible obstructions between the third and fourth row, which are not necessarily connected by identities (c.f. example ); it may be the case that and are not identities in general; however, if they are, then every morphism in the diagram is forced to be the identity as is direct; contiguous segments of are mapped to some for and for some . Then, the procedure identifies together each index labelled by the same , as they are connected by strongly collapsible morphisms. In general, the state of strong collapse after this is as follows: ["$h_0$"] 1 / "\ell ^\prime _1" <-["$h_1$"] 2 / "{\ldots }" ->["$h_{n^\prime - 2}$"] 3 / "\ell ^\prime _{n^\prime - 1}" <-["$h_{n^\prime - 1}$"] 4 / "\ell ^\prime _{n^\prime }"}; { [name=s1] 0 / "\ell ^\prime _0" ->["$h_0$"] 1 / "\ell ^\prime _1" <-["$h_1$"] 2 / "{\ldots }" ->["$h_{n^\prime - 2}$"] 3 / "\ell ^\prime _{n^\prime - 1}" <-["$h_{n^\prime - 1}$"] 4 / "\ell ^\prime _{n^\prime }"}; { [name=r1] 0 / "\ell ^\prime _0" ->["$h_0$"] 1 / "\ell ^\prime _1" <-["$h_1$"] 2 / "{\ldots }" ->["$h_{n^\prime - 2}$"] 3 / "\ell ^\prime _{n^\prime - 1}" <-["$h_{n^\prime - 1}$"] 4 / "\ell ^\prime _{n^\prime }"}; { [name=s0] 0 / "r_0" ->["$p$" above] 2 / "s_0" [xshift=2cm] <-["$q$" above] 4 / "r_1" [xshift=4cm]}; { [name=r0] 0 / "\ell _0" ->["$g_0$" below] 1 / "\ell _1" <-["$g_1$" below] 2 / "{\ldots }" ->["$g_{m^\prime - 2}$" below] 3 / "\ell _{m^\prime - 1}" <-["$g_{m^\prime - 1}$" below] 4 / "\ell _{m^\prime }."}; \foreach \x [evaluate=\x as \xp using int(\x +1)] in {1} { \foreach \y in {1, 2, 3} { (s\x \space \y ) <-[tips=false, double equal sign distance] { (r\x \space \y ), (r\xp \space \y ) }; } }; (s1 1) <-["$h_0$" above] { r2 0, r1 0 }; (s1 1) <-["$h_1$" above] { r2 2, r1 2 }; (s1 3) <-["$h_{n^\prime - 2}$" above] { r2 2, r1 2 }; (s1 3) <-["$h_{n^\prime - 1}$" above] { r2 4, r1 4 }; \foreach \x [evaluate=\x as \xp using int(\x +1)] in {0, 1} { (s\x \space 0) <-[tips=false, double equal sign distance] { (r\x \space 0), (r\xp \space 0) }; (s\x \space 4) <-[tips=false, double equal sign distance] { (r\x \space 4), (r\xp \space 4) }; }; (s0 2) <-["$b_0$" below, ""{commutative diagrams/marking, name=a}] (r1 0); (s0 2) <-["$b_1$" right] (r1 1); (s0 2) <-[""{commutative diagrams/marking, name=b}] (r1 2); (s0 2) <-["$b_{n^\prime - 1}$" right] (r1 3); (s0 2) <-["$b_{n^\prime }$" below, ""{commutative diagrams/marking, name=c}] (r1 4); (s0 2) <-["$f_0$" above, ""{commutative diagrams/marking, name=d}] (r0 0); (s0 2) <-["$f_1$" right] (r0 1); (s0 2) <-[""{commutative diagrams/marking, name=e}] (r0 2); (s0 2) <-["$f_{m^\prime - 1}$" below] (r0 3); (s0 2) <-["$f_{m^\prime }$" above, ""{commutative diagrams/marking, name=f}] (r0 4); }; \begin {scope}[on background layer] \path (a) edge[commutative diagrams/Rightarrow] (s0 0); \path (a) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (s0 4); \path (d) edge[commutative diagrams/Rightarrow] (s0 0); \path (d) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (s0 4); \end {scope} \foreach \x in {0, 1, 2, 3, 4} { \node [draw, rounded corners, fit={(r2 \x ) (r1 \x )}, inner sep=0pt] {}; \node [draw, rounded corners, fit={(r0 \x )}, inner sep=0pt] {}; } \foreach \x in {0, 2, 4} { \node [draw, rounded corners, fit={(s0 \x .center)}, inner sep=5pt] {}; } \end {tikzpicture} ]]> From here, it can be observed that is the terminal index of the strong collapse. This induces an oplax cocone over with tip for each , which then are combined to lift to the desired framed zigzag map. For the inductive case, where 1]]>, the proof essentially proceeds similarly, except this oplax cocone is obtained from the inductive hypothesis. Note that the identity cospan could have analogously been juxtaposed below instead of above, or instead have appeared within some iterated cospan, not necessarily at one of its endpoints. Nick Hu20241127https://forest.nickx.hu/coherent-inverses-5NIV/coherent-inverses-5NIV/coherent-inverses-5NIV/Juxtaposition with identity induces zigzag homotopycorollary Let , of singular height , admit some regular level , and be the framed zigzag of singular height given replacing with the identity cospan in . Then the contraction algorithm witnesses a zigzag homotopy between and . Nick Hu20241127proof We seek to witness a cospan in : Let , whence , and obtain by contraction via proposition . Nick Hu20241127https://forest.nickx.hu/coherent-inverses-61HW/coherent-inverses-61HW/coherent-inverses-61HW/Embedding is a framed zigzag maplemma Let be a framed zigzag with singular height of the following form in : ["$f_0$"] x / "\ldots " <-["$b_{k-1}$"] r_k ->["$f_k$"] s_k <-["$b_k$"] r_{k+1} ->["$f_{k+1}$"] y / "\ldots " <-["$b_{h-2}$"] r_{h-1} / "r_{h-1} ,"; }; \end {tikzpicture} ]]> satisfying (hence ). Then there is an embedding framed zigzag map whose component regular-regular and singular-singular morphisms are all identities, underlying singular monotone map is given by the -th degeneracy map, and action is to omit the cospan : ["$f_0$"] x / "\ldots " <-["$b_{k-1}$"] r_k ->["$f_k$"] s_k <-["$b_k$"] r_{k+1} ->["$f_{k+1}$"] y / "\ldots " <-["$b_{h-2}$"] r_{h-1} / "r_{h-1} ,"}; {[name=b] r_0 ->["$f_0$" below] x / "\ldots " <-["$b_{k-1}$" below] r_k [xshift=1cm] ->["$f_{k+1}$" below] y / "\ldots " [xshift=2cm] <-["$b_{h-2}$" below] r_{h-1} [xshift=2cm]}; (b r_0) ->[tips=false, double equal sign distance] (t r_0); (b r_{h-1}) ->[tips=false, double equal sign distance] (t r_{h-1}); (b x) ->[tips=false, double equal sign distance] (t x); (b y) ->[tips=false, double equal sign distance] (t y); (b r_0) ->["$f_0$" above] (t x); (b r_{h-1}) ->["$b_{h-2}$" below] (t y); (b r_k) ->[tips=false, double equal sign distance] {(t r_k), (t r_{k+1})}; (b r_k) ->["$f_k$" above right] (t s_k); (b r_k) ->["$b_{k-1}$" below] (t x); (b r_k) ->["$f_{k+1}$" below] (t y); }; \end {tikzpicture} ]]> where every 2-cell filler is trivial. Nick Hu20241127https://forest.nickx.hu/coherent-inverses-1S8Z/coherent-inverses-1S8Z/coherent-inverses-1S8Z/Juxtaposition with inverse contractsproposition Let be a diagram of the following form: ["$f$"] s1 / "s" <-["$b$"] r1 / "r^\prime " ->["$b$"] s0 / "s" <-["$f$"] r0 / "r."; }; \end {tikzpicture} ]]> Observe that this diagram corresponds the exploded diagram of a particular object of with two singular levels, where the second cospan is given by the inverse cospan of the first in . Then the contraction algorithm witnesses a framed zigzag map in whose exploded diagram is given as follows: ["$f$" left] s1 / "s" <-["$b$" left] r1 / "r^\prime " ->["$b$" left] s0 / "s" <-["$f$" left] r0 / "r"}; { [name=r, nodes={yshift=-1cm}] r1 / "r" ->["$f$"] s0 / "s" <-["$f$"] r0 / "r,"}; l r2 ->[tips=false,double equal sign distance] r r1; {l r2 [< "$f$" right], l s1 [< {tips=false, double equal sign distance}], l r1 [< "$b$" below left], l s0 [< {tips=false, double equal sign distance}], l r0 [< "$f$" right]} -> r s0; l r0 [clear <] ->[tips=false,double equal sign distance] r r0; }; \end {tikzpicture} ]]> where every 2-cell filler is trivial. Nick Hu20241127proof The proof of this is similar to that of proposition . First, for the case where , the quotient we obtain in general is: ["$f$"] s1 / "s" <-["$b$"] r1 / "r^\prime " ->["$b$"] s0 / "s" <-["$f$"] r0 / "r."; }; \foreach \x in {r2, s1, r1, s0, r0} { \node [draw, rounded corners, fit={(\x .center)}, inner sep=5pt] {}; } \end {tikzpicture} ]]> That is, the strong collapse is the same. Every component uniquely maps to a class labelled by in the strong collapse, allowing us to build a cocone with tip which can be lifted to the desired framed zigzag map (theorem ). Otherwise, when 0]]>, observe that this diagram admits a singular decomposition of into for having singular height . Each diagram explodes into a diagram of the following form in : ["$g_0$"] 1 / "l_1" <-["$g_1$"] 2 / "{\ldots }" ->["$g_{m-2}$"] 3 / "l_{m-1}" <-["$g_{m-1}$"] 4 / "l_m."}; { [name=s1] 0 / "r_0" ->["$p$" above] 2 / "s_0" [xshift=2cm] <-["$q$" above] 4 / "r_1" [xshift=4cm]}; { [name=r1] 0 / "l^\prime _0" ->["$h_0$" below right] 1 / "l^\prime _1" <-["$h_1$" below right] 2 / "{\ldots }" ->["$h_{n-2}$" below left] 3 / "l^\prime _{n-1}" <-["$h_{n-1}$" below left] 4 / "l^\prime _n"}; { [name=s0] 0 / "r_0" ->["$p$" above] 2 / "s_0" [xshift=2cm] <-["$q$" above] 4 / "r_1" [xshift=4cm]}; { [name=r0] 0 / "l_0" ->["$g_0$" below] 1 / "l_1" <-["$g_1$" below] 2 / "{\ldots }" ->["$g_{m-2}$" below] 3 / "l_{m-1}" <-["$g_{m-1}$" below] 4 / "l_m."}; \foreach \x [evaluate=\x as \xp using int(\x +1)] in {0, 1} { (s\x \space 0) <-[tips=false, double equal sign distance] { (r\x \space 0), (r\xp \space 0) }; (s\x \space 4) <-[tips=false, double equal sign distance] { (r\x \space 4), (r\xp \space 4) }; }; (s0 2) <-["$b_0$" below, ""{commutative diagrams/marking, name=a}] (r1 0); (s0 2) <-["$b_1$" right] (r1 1); (s0 2) <-[""{commutative diagrams/marking, name=b}] (r1 2); (s0 2) <-["$b_{n-1}$" right] (r1 3); (s0 2) <-["$b_n$" below, ""{commutative diagrams/marking, name=c}] (r1 4); (s0 2) <-["$f_0$" above, ""{commutative diagrams/marking, name=d}] (r0 0); (s0 2) <-["$f_1$" right] (r0 1); (s0 2) <-[""{commutative diagrams/marking, name=e}] (r0 2); (s0 2) <-["$f_{m-1}$" below] (r0 3); (s0 2) <-["$f_m$" above, ""{commutative diagrams/marking, name=f}] (r0 4); (s1 2) <-["$f_0$" below, ""{commutative diagrams/marking, name=g}] (r2 0); (s1 2) <-["$f_1$" right] (r2 1); (s1 2) <-[""{commutative diagrams/marking, name=h}] (r2 2); (s1 2) <-["$f_{n-1}$" right] (r2 3); (s1 2) <-["$f_n$" below, ""{commutative diagrams/marking, name=i}] (r2 4); (s1 2) <-["$b_0$" above, ""{commutative diagrams/marking, name=j}] (r1 0); (s1 2) <-["$b_1$" right] (r1 1); (s1 2) <-[""{commutative diagrams/marking, name=k}] (r1 2); (s1 2) <-["$b_{m-1}$" below] (r1 3); (s1 2) <-["$b_m$" above, ""{commutative diagrams/marking, name=l}] (r1 4); }; \begin {scope}[on background layer] \path (a) edge[commutative diagrams/Rightarrow] (s0 0); \path (a) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (s0 4); \path (d) edge[commutative diagrams/Rightarrow] (s0 0); \path (d) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (s0 4); \path (g) edge[commutative diagrams/Rightarrow] (s1 0); \path (g) edge[commutative diagrams/Rightarrow] (r2 1); \path (h) edge[commutative diagrams/Rightarrow] (r2 1); \path (h) edge[commutative diagrams/Rightarrow] (r2 3); \path (i) edge[commutative diagrams/Rightarrow] (r2 3); \path (i) edge[commutative diagrams/Rightarrow] (s1 4); \path (j) edge[commutative diagrams/Rightarrow] (s1 0); \path (j) edge[commutative diagrams/Rightarrow] (r1 1); \path (k) edge[commutative diagrams/Rightarrow] (r1 1); \path (k) edge[commutative diagrams/Rightarrow] (r1 3); \path (l) edge[commutative diagrams/Rightarrow] (r1 3); \path (l) edge[commutative diagrams/Rightarrow] (s1 4); \end {scope} \end {tikzpicture} ]]> Here, represents either some regular or singular level depending on whether is even or odd (the distinction here is unimportant). Now, we proceed inductively on the dimension . For the base case, where , this exploded diagram is a diagram in . Because was well-formed, so are each . contiguous segments of are mapped to some for and for some ; well-formedness in this instance allows us to deduce that the computation in the first and final rows is identical; it may be the case that and are not identities in general; however, if they are, then every morphism in the diagram is forced to be the identity as is direct; contiguous segments of are mapped to some for and for some . In general, the state of strong collapse after this is as follows: ["$g_0$"] 1 / "\ell _1" <-["$g_1$"] 2 / "{\ldots }" ->["$g_{m-2}$"] 3 / "\ell _{m-1}" <-["$g_{m-1}$"] 4 / "\ell _m."}; { [name=s1] 0 / "r_0" ->["$p$" above] 2 / "s_0" [xshift=2cm] <-["$q$" above] 4 / "r_1" [xshift=4cm]}; { [name=r1] 0 / "\ell ^\prime _0" ->["$h_0$" below right] 1 / "\ell ^\prime _1" <-["$h_1$" below right] 2 / "{\ldots }" ->["$h_{n-2}$" below left] 3 / "\ell ^\prime _{n-1}" <-["$h_{n-1}$" below left] 4 / "\ell ^\prime _n"}; { [name=s0] 0 / "r_0" ->["$p$" above] 2 / "s_0" [xshift=2cm] <-["$q$" above] 4 / "r_1" [xshift=4cm]}; { [name=r0] 0 / "\ell _0" ->["$g_0$" below] 1 / "\ell _1" <-["$g_1$" below] 2 / "{\ldots }" ->["$g_{m-2}$" below] 3 / "\ell _{m-1}" <-["$g_{m-1}$" below] 4 / "\ell _m."}; \foreach \x [evaluate=\x as \xp using int(\x +1)] in {0, 1} { (s\x \space 0) <-[tips=false, double equal sign distance] { (r\x \space 0), (r\xp \space 0) }; (s\x \space 4) <-[tips=false, double equal sign distance] { (r\x \space 4), (r\xp \space 4) }; }; (s0 2) <-["$b_0$" below, ""{commutative diagrams/marking, name=a}] (r1 0); (s0 2) <-["$b_1$" right] (r1 1); (s0 2) <-[""{commutative diagrams/marking, name=b}] (r1 2); (s0 2) <-["$b_{n-1}$" right] (r1 3); (s0 2) <-["$b_n$" below, ""{commutative diagrams/marking, name=c}] (r1 4); (s0 2) <-["$f_0$" above, ""{commutative diagrams/marking, name=d}] (r0 0); (s0 2) <-["$f_1$" right] (r0 1); (s0 2) <-[""{commutative diagrams/marking, name=e}] (r0 2); (s0 2) <-["$f_{m-1}$" below] (r0 3); (s0 2) <-["$f_m$" above, ""{commutative diagrams/marking, name=f}] (r0 4); (s1 2) <-["$f_0$" below, ""{commutative diagrams/marking, name=g}] (r2 0); (s1 2) <-["$f_1$" right] (r2 1); (s1 2) <-[""{commutative diagrams/marking, name=h}] (r2 2); (s1 2) <-["$f_{n-1}$" right] (r2 3); (s1 2) <-["$f_n$" below, ""{commutative diagrams/marking, name=i}] (r2 4); (s1 2) <-["$b_0$" above, ""{commutative diagrams/marking, name=j}] (r1 0); (s1 2) <-["$b_1$" right] (r1 1); (s1 2) <-[""{commutative diagrams/marking, name=k}] (r1 2); (s1 2) <-["$b_{m-1}$" below] (r1 3); (s1 2) <-["$b_m$" above, ""{commutative diagrams/marking, name=l}] (r1 4); }; \begin {scope}[on background layer] \path (a) edge[commutative diagrams/Rightarrow] (s0 0); \path (a) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 1); \path (b) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (r1 3); \path (c) edge[commutative diagrams/Rightarrow] (s0 4); \path (d) edge[commutative diagrams/Rightarrow] (s0 0); \path (d) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 1); \path (e) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (r0 3); \path (f) edge[commutative diagrams/Rightarrow] (s0 4); \path (g) edge[commutative diagrams/Rightarrow] (s1 0); \path (g) edge[commutative diagrams/Rightarrow] (r2 1); \path (h) edge[commutative diagrams/Rightarrow] (r2 1); \path (h) edge[commutative diagrams/Rightarrow] (r2 3); \path (i) edge[commutative diagrams/Rightarrow] (r2 3); \path (i) edge[commutative diagrams/Rightarrow] (s1 4); \path (j) edge[commutative diagrams/Rightarrow] (s1 0); \path (j) edge[commutative diagrams/Rightarrow] (r1 1); \path (k) edge[commutative diagrams/Rightarrow] (r1 1); \path (k) edge[commutative diagrams/Rightarrow] (r1 3); \path (l) edge[commutative diagrams/Rightarrow] (r1 3); \path (l) edge[commutative diagrams/Rightarrow] (s1 4); \end {scope} \foreach \x in {0, 1} { \foreach \y in {0, 2, 4} { \node [draw, rounded corners, fit={(s\x \space \y )}, inner sep=0pt] {}; } } \foreach \x in {0, 1, 2} { \foreach \y in {0, 1, 2, 3, 4} { \node [draw, rounded corners, fit={(r\x \space \y )}, inner sep=0pt] {}; } } \end {tikzpicture} ]]> As before, this witnesses two oplax cocones with tip : one over the restriction of this diagram to the first three rows, and the other over the restriction to the bottom three rows. The fact that there is symmetry in the middle three rows, arising from that fragment of the diagram originating from the exploded diagram of (some part of the singular decomposition of) ensures that these two oplax cocones are compatible, i.e. they agree on the common middle row. This means that both oplax cocones can be glued to form an oplax cocone over the entire diagram. Repeating this for each to obtain a sequence of oplax cocones with tip for each , we lift the result into the desired framed zigzag map. As before, when 1]]>, the remainder of the proof is given inductively. Nick Hu20241127https://forest.nickx.hu/coherent-inverses-XBVP/coherent-inverses-XBVP/coherent-inverses-XBVP/Juxtaposition with inverse induces zigzag homotopycorollary Let , of singular height , admit some regular level , and be some cospan in . Moreover, let be framed zigzag of singular height obtained by replacing with in . Then the contraction algorithm witnesses a zigzag homotopy between and . Nick Hu20241127proof We seek to witness a cospan in : Obtain via lemma , whence is the framed zigzag of singular height which is given by but with an additional cospan in place of , and obtain by contraction via proposition . It is by this mechanism that contraction algorithm is able to construct higher-dimensional coherences for invertibility, as in figure . Nick Hu2025117https://forest.nickx.hu/coherent-inverses-O844/coherent-inverses-O844/coherent-inverses-O844/Constructing a homotopy cap by contractionfigure Constructing the 2-cell by contraction.Nick Hu2023123https://forest.nickx.hu/coherent-inverses-implementation/coherent-inverses-implementation/coherent-inverses-implementation/Implementationchapter In this chapter, we discuss some of the implementation details of homotopy.io. The main reference for this chapter is [homotopy-io-a-proof-assistant-for-finitely-presented-globular-n-categories]. Nick Hu202518https://forest.nickx.hu/coherent-inverses-JGBQ/coherent-inverses-JGBQ/coherent-inverses-JGBQ/homotopy.io interfacefigure The interface of homotopy.io.homotopy.io is written in the Rust programming language, targeting the web browser via WebAssembly. This has consequences for our implementation, as we must choose an encoding that is expressible within the Rust type system, and our target platform also restricts the kind of operations we can perform and places constraints on resources. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-implementation-encoding/coherent-inverses-implementation-encoding/coherent-inverses-implementation-encoding/Encoding framed zigzags and framed zigzag mapssection We will present a simplified version of the algebraic data types used in homotopy.io to encode the theory of framed zigzags, in pseudo-ML syntax. In code, a framed zigzag is called a diagram, while a framed zigzag map is called a rewrite. Nick Hu2024124https://forest.nickx.hu/coherent-inverses-6O54/coherent-inverses-6O54/coherent-inverses-6O54/Data structures of homotopy.iocode Here, we represent framing with integers. The generator type is essentially a name equipped with some metadata, and the signature type, encoding an algebraic signature, is a mapping of these names to their canonical diagrams — every generator has an associated diagram, but the converse is not necessarily true. We use a lightweight form of dependent typing to separate 0-dimensional framed zigzags and framed zigzag maps from -dimensional ones (where 0]]>). Although not explicit, each DiagramN and RewriteN has an associated dimension 0]]>, and should maintain the invariant that its component pieces are made up of only Diagram0 and Rewrite0 if , or DiagramN and RewriteN all of associated dimension otherwise. This choice of encoding has the following limitation: there are unenforced invariants that allow for valid terms of the diagram type and do not correspond to an admissible -cell of the free higher category generated by any signature. For example, a Diagram0 is supposed to represent a 0-cell, but when its generator has a dimension field not set to 0 this will not be the case. As is always the case with programming languages without proper dependent typing, the responsibility for maintaining these invariants falls on the programmer. The other non-trivial feature of this encoding is the way framed zigzags and framed zigzag maps are encoded sparsely. Suppose that . is some iterated cospan in of singular height ; a compact way of representing this is to store only the distinguished source object of , along with all the morphisms of within this iterated cospan; i.e. we store the coloured parts of from which the rest of the data can be reconstructed by auxiliary procedures that allow for diagrams to be rewritten both forwards and backwards along rewrites of the appropriate associated dimension. Note that the singular height of a (-dimensional, where 0]]>) diagram is given by the number of cospans that it contains. In this encoding, we think of the rewrites (which encode framed zigzag maps) as the differences between adjacent regular/singular slice in the diagram, where each set of differences is given by a sequence of contiguously acting differences, which we call cones. To expand on this, for framed zigzag maps in , rather than storing their entire underlying monotone map, along with the accompanying morphisms in , we only store the parts that deviate from the identity (locally) in the form of cones. This is analogous to how large numerical matrices are stored in a sparse format, where only the non-zero entries are stored. In practice, this makes for significant savings in space efficiency, because most adjacent heights in -dimensional string diagrams only differ in small parts. The following example demonstrates this encoding of a framed zigzag in of singular height 1. Nick Hu2024124https://forest.nickx.hu/coherent-inverses-2C1T/coherent-inverses-2C1T/coherent-inverses-2C1T/framed zigzag maps in conesexample Consider the following 2D string diagram: This diagram is made up of a DiagramN whose source is a DiagramN encoding the bottom (horizontal) slice 1-dimensional string diagram, and whose cospans are given by the singleton list with a sole cospan whose forward field comprises a RewriteN whose cones are given by and , and backward field comprises a RewriteN whose cones are given by and . and are identity cones; the fragment of the framed zigzag map local to where they act is the identity framed zigzag map. This is why the cone type has an index field: we omit and explicitly from the cones list in their containing RewriteN, but the cones and will have index 1, meaning that there is one identity cone to their left. The following diagram shows the bottom half of the figure above more explicitly, as a framed zigzag map: In some sense, we can see a framed zigzag map as a modification of the sequence of cospans in by replacement of subsequences by individual cospans of . Each such modification is a cone, and here the sequence of cospans is replaced by . Nick Hu2024124https://forest.nickx.hu/coherent-inverses-HV2T/coherent-inverses-HV2T/coherent-inverses-HV2T/Identity -rewritesremark The encoding of an identity framed zigzag map in is given by RewriteN with the empty list of cones. Much of the technical complexity of the implementation of homotopy.io is about phrasing our procedures with respect to this sparse cone representation and maintaining the associated invariants. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-implementation-memoisation/coherent-inverses-implementation-memoisation/coherent-inverses-implementation-memoisation/Memoisationsection The data structures outlined in code are immutable. Operations that would modify an object instead create a new instance of the object with the desired modifications. We do this so that the technique of hash consing can be applied: we maintain a global hash table that tracks all the objects that are resident in memory, and then whenever a new object is created we first check against this global table to reuse an existing allocation if possible (the soundness of this technique is predicated on the immutability of our data structures). Not only does this save memory, but an additional effect is that pointer equality becomes sufficient to determine structural equality, which makes equality comparisons of diagrams and rewrites complete in time. As an extension, due to the highly mutually recursive relationship between diagrams and rewrites, we also cache their hash values to avoid deep traversals. Empirically, this technique has been effective at reducing memory usage and improving performance; even with our sparse encoding, large high-dimensional diagrams are still highly redundant in practice. The key algorithms of homotopy.io (chapter ) are largely recursive over the structure of diagrams and rewrites, performing many identical recursive calls on logically equal substructures. We can exploit this by using memoisation to cache the results of these recursive calls and avoid redundant computation. Due to hash consing, we have very fast equality checking, so lookups in the memoisation table are comparatively cheap. The global hash consing table counts references to all the live objects, and when an object is no longer needed (when its reference count is zero), it can be safely removed from the table. However, the nature of our algorithms will cause the same diagrams and rewrites to be created and destroyed many times over the course of a computation, so we deliberately avoid an eager approach to garbage collection to avoid memory churn, only performing garbage collection when the program is suitably idle. This technique also enables the serialisation/deserialisation of diagrams and rewrites, which otherwise, being mutually recursive, forms a cyclic data structure. We use this to save and load data in a binary format via MessagePack, allowing for a user to persist their work across sessions or share it with others. On deserialisation, we sanitise the input data via our typechecking algorithm to ensure that it was mathematically valid. This mechanism is extended by an arXiv-style server implementation, with which the proof assistant communicates as a client, where users can upload their work (called a project) to a central server and share it by a link, such as https://beta.homotopy.io/p/2402.00001 (workspace for the associator 3-diagram). Publishing a project versions and timestamps it, making it read-only, producing a unique identifier like an arXiv slug that can be used to access the project at any time. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-implementation-rendering/coherent-inverses-implementation-rendering/coherent-inverses-implementation-rendering/Rendering string diagramssectionhomotopy.io, as a graphical proof assistant for -dimensional string diagrams, must provide a way to render these diagrams according to standard conventions in the field. That is, a 2D string diagram should be recognisable as such, with the 2-morphisms represented by points, the 1-morphisms by lines, and the 0-morphisms by dots. To accomplish this, we have procedures that take the combinatorial data, presented as diagrams and rewrites, and render them graphically as SVG images (for 2D viewing) or via WebGL surfaces/animations (for 3D/4D viewing) with a camera control mechanism. Nick Hu202518https://forest.nickx.hu/coherent-inverses-QR5Z/coherent-inverses-QR5Z/coherent-inverses-QR5Z/Monoid 2D string diagramfigure 2D string diagram of a monoid. We sketch this process, deferring a more detailed explanation to [§ 6.2, homotopy-io-a-proof-assistant-for-finitely-presented-globular-n-categories]. The theory of the layout algorithm, which is interesting in its own right from a categorical perspective, can be found in [a-layout-algorithm-for-higher-dimensional-string-diagrams] and [a-computational-approach-to-higher-categories]. The issue is that geometry in dimensions higher than two is difficult to visualise directly when the expected mode of interaction is a screen. We use projections to overcome this: -dimensional space can be projected down to -dimensional space, like how physically 3D objects cast 2D shadows. For 3D, we can render the 3D surface diagram as 3D geometry within a virtual 3D space, and then present a viewport of a camera within this space to the user. The camera sees a projection of the 3D space onto a 2D plane, which can be displayed on a screen. By giving the user control over the camera (e.g. by allowing them to pan and rotate the camera around the 3D space), we can give the user a sense of the 3D structure of the diagram. This is the same principle as how 3D games are rendered on a 2D screen. Lighting and shading, along with perspective projection, are used to create a more realistic 3D effect. Nick Hu202518https://forest.nickx.hu/coherent-inverses-YQ56/coherent-inverses-YQ56/coherent-inverses-YQ56/Monoid associator 3D surface diagramfigure Monoid associator 3D surface diagram, rendered with camera angle at 0°, 30°, 60°, and 90°. Similarly to 2D string diagrams, cells with dimension 3 in the projected 3D space (codimension 0) are rendered as points, cells with dimension 2 (codimension 1) are rendered as wires, and cells with dimension 1 (codimension 2) are rendered as surfaces. Each delimited volume in the 3D space also notionally corresponds to a cell with dimension 0 (codimension 3), but these are deliberately elided by our renderer for clarity. For 4D geometry, we can represent the additional dimension as time, by constructing an animation of 3D projections of the 4D geometry evolving over time (also equipped with camera control as above). The smoothness of the animation, achieved via subdivision, is crucial to communicating the structure of the 4D diagram. Nick Hu202518https://forest.nickx.hu/coherent-inverses-BMQ3/coherent-inverses-BMQ3/coherent-inverses-BMQ3/Swallowtail coherence 4D animated surface diagramfigure Swallowtail coherence 4D diagram, rendered at time . This process can also be performed to present a 3D diagram as an animation of 2D projections by interpreting the Z-axis as time. Via the smooth subdivision, this obtains a continuous deformation that interpolates between regular and singular slices, which alludes to why the tool is called homotopy.io. Nick Hu202518https://forest.nickx.hu/coherent-inverses-4I1C/coherent-inverses-4I1C/coherent-inverses-4I1C/Monoid associator 3D animated string diagramfigure The associator 3D diagram, rendered as 2D diagrams at time . Our renderer has many different output formats, including SVG and TikZ, which is used heavily to generate the high quality string diagram pictures in this thesis and other papers. In 3D, we can also output to the STL file format for 3D printing. Nick Hu202518https://forest.nickx.hu/coherent-inverses-0MD5/coherent-inverses-0MD5/coherent-inverses-0MD5/Monoid associator 3D print of surface diagramfigure 3D print of the associator surface diagram.Nick Hu2023123https://forest.nickx.hu/coherent-inverses-implementation-morse-projection/coherent-inverses-implementation-morse-projection/coherent-inverses-implementation-morse-projection/The ‘Morse projection’ of -dimensional string diagramssection We have discussed how to render -dimensional string diagrams in 2D, 3D, and 4D for when , but in general we would like our tool to work in arbitrary dimensions. To achieve this, we use a different notion of projection to (lossfully) reduce the dimension of the diagram, which we call the ‘Morse projection’. This projection operation is built in to the structure of framed zigzags, in that objects in an iterated framed zigzag enriched category come with a natural sequence of diagrams, which we interpret geometrically, via their explosion bundle. In some sense, explosion is a dual operation to projection. Assume that . In the first level of explosion, we obtain the iterated cospan of that presents , which we can draw vertically as a 1D diagram — i.e., we have interpreted the iterated cospan geometrically along the Y-axis. Now, in the second level of explosion, each of the indices in the 1D diagram itself corresponds to some iterated cospan in , and the framed zigzag maps in also explode to create framed zigzag maps in that intervene between each level; we interpret this second level of explosion geometrically along the X-axis. Nick Hu2024125https://forest.nickx.hu/coherent-inverses-MSZR/coherent-inverses-MSZR/coherent-inverses-MSZR/Explosion of a monoid as an object example Let be the signature consisting of a single 0-cell , a 1-cell , and a 2-cell (representing a monoid over ). There is an object , associated to , which represents the combinatorial encoding of the monoid 2D string diagram. The following picture shows the explosion of this object (framing labelling has been elided for clarity); note that the fully exploded diagram in looks like a ‘wireframe’ of a monoid string diagram. f <- x }; { [name=s0, fresh nodes] x -> m [orange] <- x }; { [name=r0, fresh nodes, nodes={xshift=-1cm}] x -> f <- x -> f <- x }; { [edges=gray] r1 x -> s0 m <- r0 x; r1 x' -> s0 m <- r0 x''; r0 x' -> s0 m; }; r1 x -> s0 x <- r0 x; r1 x' -> s0 x' <- r0 x''; { [edges=red] s0 m <- { r0 f, r0 f', r1 f }; }; }; \end {tikzpicture} \ar [r, leftsquigarrow, "\text {explode}", shorten >=2pt, shorten <=2pt] & \begin {tikzpicture}[baseline=(current bounding box.center), every node/.append style={draw=none}] \node [draw=gray] [yshift=0.5cm] {$\operatorname {Zig}(\Sigma )$}; \path graph[math nodes] { T; C; S; T ->["$b$"] C <-["$f$"] S; }; \end {tikzpicture} \ar [r, leftsquigarrow, "\text {explode}", shorten >=2pt, shorten <=2pt] & \begin {tikzpicture} \node [draw=none] (Z) {$Z$}; \node [above=0.25cm of Z] (c) {$\operatorname {Zig}^{2}(\Sigma )$}; \end {tikzpicture} \end {tikzcd} ]]> Recall that, in our representation, -dimensional diagrams are fundamentally given by a sequence of -dimensional diagrams assembled into a collection of regular and singular slices. This means that, so long as is sufficiently large (e.g. ), we can render any -dimensional diagram as a 2D/3D/4D diagram by projection to the appropriate codimension. Topologically, this is something like a ‘Morse projection’ of the diagram, which we illustrate in the following example. Nick Hu2024125https://forest.nickx.hu/coherent-inverses-L5HR/coherent-inverses-L5HR/coherent-inverses-L5HR/Morse projection of a monoid multiplicationexample A monoid multiplication as a 2D string diagram can be rendered as a 1D diagram by projecting out the X-axis. Subsequently, we can obtain a 0D diagram by projecting out the Y-axis (which always looks like a point). In pictures, this looks like the following: ] (-1,2) -- node[below, yshift=-0.2cm] {\small project} (-2,2); \end{scope} \begin{scope}[xshift=2 cm,plane origin={(0,0,0)}, plane x={(0,1,1)}, plane y={(0,1,0)}, canvas is plane] \begin{scope} \clip (0,0) -- (4,0) -- (4,4) -- (0,4) -- (0,0); \draw[color=generator-1-1-0-pos, line width=4pt](2,-1) -- (2,5); \fill[generator-2-2-0-pos] (2,2) circle (0.14); \end{scope} \draw[->] (2,-0.4) -- node[right] {\small project} (2,-1.8); \end{scope} \begin{scope}[xshift=-1 cm, canvas is xz plane at y=0] \fill[generator-2-2-0-pos] (3,2) circle (0.14); \end{scope} \end {tikzpicture} ]]> Another feature of this representation is that we can specify a sub--dimensional diagram within an -dimensional diagram, for by giving a -length sequence of slice heights. View control, determined by the following regular expression: combines this information with the desired render dimension , where here and are special indices that refer to the ‘source’ (first) regular height and the ‘target’ (last) regular heights respectively ​ . A word in this language determines, for an -dimensional diagram, that our view is the -dimensional diagram specified by the first symbols, being viewed in dimensions, projecting the remaining dimensions. We present each symbol as a button that can be clicked to change the view: clicking sets (setting the view to the original -diagram); clicking any button decrements by 1, to a minimum of 0 (rendered as a single point), and clicking any button increments by 1, to a maximum of 4. Descending into a slice (increasing ) is given by a sequence of chevron buttons ▸, one for each regular and singular height of the diagram and also for the source and target, aligned vertically to the right of the diagram. Even in 3D and 4D string diagrams, the primary mode of interaction is via a 2D slices and projections, with the 3D and 4D renderers mainly for visualisation. Now that we have established which parts of a string diagram are to be rendered, to produce an image on screen, we proceed via the following pipeline: layout — assign real-valued coordinates in -dimensional Euclidean space , for depending on the type of render requested, to each piece of the diagram: [a-layout-algorithm-for-higher-dimensional-string-diagrams] describes this process as a categorical construction which extracts a set of linear constraints from the diagram; these are then augmented with additional aesthetic constraints (e.g. ‘try to centre wires and surfaces’), before being passed to the HiGHS linear programming solver; mesh generation — compute a representation of this geometry as a collection of -dimensional cubes, for ; subdivision — refine the mesh, making it smoother, by recursively subdividing each cube into smaller cubes. Due to our target platform being the web browser, one limitation inherited from WebGL is that there is no access to compute shaders (loosely speaking, these allow for general-purpose compute on the GPU), which would be the natural way to compute the animation in the 4D case. Instead, a 4D mesh is triangulated using a rendering trick [§ 6.2, homotopy-io-a-proof-assistant-for-finitely-presented-globular-n-categories] so that it can be rendered as if it were a 3D mesh, which aligns to the capabilities of WebGL. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-implementation-architecture/coherent-inverses-implementation-architecture/coherent-inverses-implementation-architecture/System architecturesectionhomotopy.io as a complex system is composed of many interacting parts. The codebase is split into many different ‘crates’ (Rust's notion of package), which themselves are split into many different modules. The main algorithmic core is contained in the homotopy-core crate, which has modules such as diagram.rs and rewrite.rs, that define some of the algebraic data types of code and implement auxiliary methods to support them. collapse.rs and contraction.rs contain the main algorithms of homotopy.io, as described in section . Aside from this, the system is broadly architected using the Redux pattern (see: homotopy-model), popularised by the React.js and Redux JavaScript libraries. There is a single global store containing the current state of the system, which includes not only the mathematical state (e.g. the signature, the diagram being interacted with in the workspace), but also the ancillary state (e.g. which sidebar is open, pan/zoom transformations). Every action that can be performed in the system is represented as data in the form of a Rust enum. Now, instead of directly modifying the state in response to user interaction, we instead signal an action to the system, where it is handled by a state reducer that computes the new state from the old state and the action. The view, which is presented to the user, is computed as a pure function of the state, updating whenever the state changes. Each state can be described as a sequence of all the actions that have been applied to the initial state, which is useful for debugging and for undo/redo functionality. Additionally, actions can be filtered for validity based on the current state by the reducer, which can be used to disable UI elements that do not make sense in the current context (e.g. one cannot slice a 0D diagram). This leads to a strict decoupling of the UI from the underlying logic, which is a key design principle of the system. As a result of this modularity, we are able to support two seperate interfaces: a web interface (homotopy-web), which is the main one, and a command line interface (homotopy-cli) for debugging. Moreover, this separation of concerns allows the pieces of the system to be individually tested; many of the technicalities of our algorithms and data structures are elaborate, even in their atomic parts, so it is all but necessary to be able to apply a rigorous testing methodology to validate correctness. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-implementation-interaction/coherent-inverses-implementation-interaction/coherent-inverses-implementation-interaction/Interacting with the systemsection There are two main mathematically significant ways in which the user can interact with homotopy.io: generator creation The user can always click a button to add a new (fresh) 0-cell to the signature; new -cells, for 0]]>, are created by constructing their source and target boundary, satisfying globularity, and designating them as ‘source’ and ‘target’ with the dedicated buttons to do so. These correspond to the functions in the generator creation algorithm. Additionally, generators in the signature can be marked as invertible, making them eligible for attachment in their inverse orientation. Any finite presentation of a higher globular -category (or -groupoid) can be input in this way. homotopy construction Given some 2D (or 1D) diagram, rendered on screen in the workspace, the user can interact with it by clicking on certain regions to attach cells from the signature to the diagram, or by clicking-and-dragging to trigger contraction and expansion moves. The system responds to these interactions by computing the appropriate diagrams and rewrites and updating the workspace, or if the move is invalid, by displaying an error message. If a contraction/expansion is performed within some subdiagram of a larger diagram, there are auxiliary algorithms that propagate the change to the entire diagram [§ 5, high-level-methods-for-homotopy-construction-in-associative-n-categories]. Homotopy construction is facilitated by two main procedures: attachment, and contraction alongside its dual operation expansion, which we briefly describe here. Nick Hu2024125https://forest.nickx.hu/coherent-inverses-TQBG/coherent-inverses-TQBG/coherent-inverses-TQBG/Expansionsection Expansion is dual to the contraction algorithm, first described by [§ 5.2, high-level-methods-for-homotopy-construction-in-associative-n-categories] and later refined by [the-theory-and-applications-of-anticolimits], [Chapter 6, a-computational-approach-to-higher-categories]. It is an inductive procedure which, given some diagram of singular height , produces an expanded diagram of singular height along with a rewrite such that the contraction algorithm on at succeeds, returning as its output, or fails. In this sense, expansion is a partial (non-unique) converse to contraction. To provide the mouse interface, each 2D string diagram image (which is rendered as an SVG) is composed as a collection of triangles; then, certain triangles are equipped with event listeners to trigger the appropriate action. This is best described by example. Nick Hu2024127https://forest.nickx.hu/coherent-inverses-18UO/coherent-inverses-18UO/coherent-inverses-18UO/Attachment, contraction, expansionexample Fix a signature with a 0-cell and an invertible scalar 2-cell . As 2D string diagrams, and its inverse look like the following: The diagram for admits two attachment points, at the source and target boundaries: Clicking in the ‘target’ region triggers an attachment procedure which works as follows: enumerate the valid diagrams from the signature which can be attached here; if there is more than one valid diagram, prompt the user to choose one; attach the chosen diagram to this attachment point and update. In this case, because and are valid 2-cells in this -category, the user is prompted to choose between attaching or . If the user chooses to attach , the diagram (now corresponding to ) becomes the following: From here, the following sequence of click-and-drags (starting on ) triggers a contraction followed by an expansion: , thick, yellow, shorten <=12px, opacity=0.4] (2,4) -- node[midway, right] {drag} (3,2.5); \end{tikzpicture} }; \node[right=2cm of a] (b) { \begin{tikzpicture}[baseline=(current bounding box.center)] \begin{scope} % Background surfaces \fill[generator-0-0-0-pos] (0,0) -- (6,0) -- (6,4) -- (0,4) -- (0,0); % Wire layers \end{scope} \fill[generator-1-2-0-pos] (2,2) circle (0.14); \fill[generator-2-2-0-pos] (4,2) circle (0.14); \draw[->, thick, yellow, shorten <=12px, opacity=0.4] (4,2) -- node[midway, right] {drag} (3,0.5); \end{tikzpicture} }; \node[right=2cm of b] (c) { \begin{tikzpicture}[baseline=(current bounding box.center)] \begin{scope} % Background surfaces \fill[generator-0-0-0-pos] (0,0) -- (4,0) -- (4,6) -- (0,6) -- (0,0); % Wire layers \end{scope} \fill[generator-2-2-0-pos] (2,2) circle (0.14); \fill[generator-1-2-0-pos] (2,4) circle (0.14); \end{tikzpicture} }; \draw (a) edge[commutative diagrams/rightsquigarrow] node[above] {contraction} (b) (b) edge[commutative diagrams/rightsquigarrow] node[above] {expansion} (c); \end {tikzpicture} ]]> The homotopy that has been constructed here can be interpreted as the Eckmann-Hilton argument: in a 3-category, the two ways of composing the two 2-cells are equal. Geometrically, we can interpret this as a braid, tracing the wordlines of the 2-cells; this is also the crux of the idea that braided monoidal categories are equivalent to doubly-degenerate 3-categories (a single 0-cell, no non-identity 1-cells): Indeed, figure shows what the 3D renderer draws for this 3-cell. Nick Hu202519https://forest.nickx.hu/coherent-inverses-ZKW8/coherent-inverses-ZKW8/coherent-inverses-ZKW8/Braid 3D tube diagramfigure Eckmann-Hilton braid of a scalar with its inverse. If instead for the first contraction we had dragged the other way, we could have created the other braid. Moreover, dragging directly vertically downwards triggers a third mode of contraction whereby we cancel with : , thick, yellow, shorten <=12px, opacity=0.4] (2,4) -- node[midway, right] {drag} (2,2.5); \end{tikzpicture} }; \node[right=2cm of a] (b) { \begin{tikzpicture} \begin{scope} % Background surfaces \fill[generator-0-0-0-pos] (0,0) -- (2,0) -- (2,2) -- (0,2) -- (0,0); % Wire layers \end{scope} \end{tikzpicture} }; \draw (a) edge[commutative diagrams/rightsquigarrow] node[above] {contraction} (b); \end {tikzpicture} ]]> This kind of mode can be forced by holding the ⇧ Shift key while dragging. We could have also dragged the point instead of the point to construct these homotopies symmetrically. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-applications/coherent-inverses-applications/coherent-inverses-applications/Applicationschapter In this chapter, we explore some of the applications of homotopy.io as a proof assistant, with a particular focus on proofs that feature the use of higher-dimensional coherences arising from invertibility. We will present a zigzag homotopy as some Morse projection to 2-dimensional string diagrams; in some sense, this is a proof-relevant version (see next section) of equations between string diagrams in the ordinary diagrammatic calculus, where the legs and tips of each cospan is the singularity equipped with homotopies that witnesses the homotopical information underlying each equality (allowing for this whole zigzag homotopy to be interpreted as a higher cell). The proof-irrelevant notion can be recovered by ignoring every even-numbered diagram within a zigzag homotopy (i.e. only looking at the feet of its underlying iterated cospan). We elide the information contained within each framed zigzag map, but provide links to online proofs in homotopy.io that allow for an interactive exploration of these higher cells. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-applications-equations/coherent-inverses-applications-equations/coherent-inverses-applications-equations/Invertible cells as proof-relevant equationssection It is well-known that every monoidal category is equivalent to a degenerate (by which we mean that there is exactly one object) bicategory, analogously to how a monoid is equivalent to a one-object category. This notion extends further: every braided monoidal category is equivalent to a doubly degenerate tricategory, and every symmetric monoidal category is equivalent to a triply degenerate tetracategory. In fact, the generalisation of this phenomenon to -categories, the delooping hypothesis, for the ‘Periodic Table of -categories’ [Figure 2, the-periodic-table-of-n-categories-for-low-dimensions-i-degenerate-categories-and-degenerate-bicategories] [§ 5, higher-dimensional-algebra-and-topological-quantum-field-theory], allows us to encode the respective string diagrammatic calculi for planar, braided, and symmetric monoidal categories in homotopy.io as special cases of its -dimensional string diagrammatic calculus under a 2-dimensional Morse projection by choosing an appropriate signature with only one 0-cell (which is interpreted as the ‘ambient space’) and no 1/2-cells (hence ensuring doubly/triply degenerateness: the unique 1-cells and 2-cells are then given by the identity on the ambient space, and the identity on the identity on the ambient space respectively). The equalities from the string diagrammatic calculus of planar/braided/symmetric monoidal categories are proof-irrelevant, in the sense that a well-typed (satisfying globularity) equation between string diagrams either holds or does not. In homotopy.io, these are instead represented as (not necessarily unique) 3/4/5-cells that then witness the equality as a higher cell. As equations, it is natural to expect these cells to be coherently invertible. A severe limitation of the previous iteration of the tool, which did not support invertible generators of any form, was that this desired behaviour could only be emulated by manually inputting each equation as a directed equality, and then furthermore any higher coherences would also be subject to manual input. For example, for some equality of string diagrams , the user would have to manually input a cell witnessing , and then its symmetric (inverse) cell ; then, if the user wished to use the fact that rewriting along should be equivalent to the ‘do-nothing’ rewrite, that data then itself needs to be input as a cell of one dimension higher, and so on ad infinitum. Our new version rectifies this issue. We exemplify this with a workspace containing a Frobenius algebra, in the planar string diagrammatic calculus viewed this way. Letting be the 0-cell representing the ambient space, and be the 1-cell representing the carrier (the ‘wire’ type) for the Frobenius algebra, we can construct the following 2-cells: each of which represents the planar string diagram associated to the monoid multiplication, unit, comultiplication, and counit respectively. Equations that witness the algebraic structure, e.g. that the monoid multiplication is associative, that the monoid and comonoid interact to witness the Frobenius laws, etc., are then represented as invertible 3-cells. All in all, this signature as a whole is presented in homotopy.io as the figure (using folders to group the equations). Nick Hu202518https://forest.nickx.hu/coherent-inverses-7WXL/coherent-inverses-7WXL/coherent-inverses-7WXL/Frobenius signaturefigure homotopy.io signature for the Frobenius algebra. For instance, using the fact that the equations are represented by invertible cells, it is possible to derive the Frobenius equation: despite this not being explicitly represented in the signature, via the composite zigzag homotopy: The first zigzag homotopy is the first Frobenius law, and the second zigzag homotopy is the inverse of the second Frobenius law. Link to online proof: https://beta.homotopy.io/p/2412.00003. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-applications-catastrophes/coherent-inverses-applications-catastrophes/coherent-inverses-applications-catastrophes/Higher-dimensional catastrophes arising from a dualisable 1-morphismsection For this section, we consider the signature containing two 0-cells, and , and a single dualisable 1-cell . Such a signature presents a theory suited to the description of higher-categorical analogues of adjunction, given that a dualisable 1-cell in a 2-category is a straightforward generalisation of the concept of adjoint functors (taking 1-cells to be functors in the 2-category , where a choice of duals is given by choosing a left/right adjoint for a functor). In homotopy.io, we emulate the dualisability of by making it invertible and restricting ourselves to only using the snake equations of figure as opposed to all the 3-cells arising from the coherent invertibility of , as invertibility implies dualisability. Informally, this is justified by the fact that every equivalence can be viewed as an adjoint equivalence, which is conjecturally expected to generalise to arbitrary dimensions. The coherence of this invertibility then suffices to generate the coherences expected for so-called ‘full dualisability’, i.e. that the structure maps of the duality unit and counit also admit adjoints in the dimension above, and so on up to arbitrary -dimensions. For instance, the invertible admits the invertible 3-cells as in figure . If were dualisable but not invertible, then the 3-cells in the first row would be missing. We expand on this in the following section. Nick Hu202518https://forest.nickx.hu/coherent-inverses-G29R/coherent-inverses-G29R/coherent-inverses-G29R/Invertible 1-morphism coherencesfigure Coherence 3-cells arising from an invertible 1-morphism. We demonstrate that homotopy.io can automatically derive some of these higher-dimensional coherences, which we call ‘catastrophes’ (borrowing terminology from Catastrophe Theory). Nick Hu2023123https://forest.nickx.hu/coherent-inverses-applications-catastrophes-snake/coherent-inverses-applications-catastrophes-snake/coherent-inverses-applications-catastrophes-snake/Snake coherencesection A snake coherence on is a 3-dimensional coherence that asserts that the zigzag homotopy: is equivalent to the identity 2-cell. Interpreted equationally, this says that the sequence of transformations where each one-directional arrow represents a 2-cell arising from admitting as a dual (adjoint), which has type is equivalent to the ‘do-nothing’ transformation . This is a generalisation of one of the so-called ‘triangle identities’ in the context of adjoint functors, which when drawn as 2-dimensional string diagrams yields a ‘straightening of the snake’ picture. In this sense, the coherence data is precisely the witness that the snake can be straightened. Such a coherence is given by the following zigzag homotopy, constructed by contraction: Viewed as a 3-dimensional string diagram, this zigzag homotopy looks like figure . Note that this contrasts with the figure in example : it is a different snake (there are two associated to , and two associated to ), and more importantly it is homotopical rather than algebraic — there is no purple dot representing a new algebraic generator in the middle of the diagram, and its zigzag homotopy has been generated entirely by contraction. Nick Hu202518https://forest.nickx.hu/coherent-inverses-21R2/coherent-inverses-21R2/coherent-inverses-21R2/Snake coherencefigure Snake coherence 3D surface diagram.Nick Hu2023123https://forest.nickx.hu/coherent-inverses-applications-catastrophes-swallowtail/coherent-inverses-applications-catastrophes-swallowtail/coherent-inverses-applications-catastrophes-swallowtail/Swallowtail coherencesection A swallowtail coherence on the cup arises as a 4-dimensional coherence on a snake, which explicitly is given by the assertion that the zigzag homotopy: is equivalent to the identity 3-cell. Starting from the second regular slice (the third image above), there are two ways to reach the cup: perform a snake cancellation move on the left; interchange the cups, to obtain the third regular slice, and then perform a snake cancellation move on the right. In this sense, the swallowtail is a coherence for the snake, as it is a higher-dimensional cell that witnesses that these two ways of reaching the cup cancel out. Equivalently, figure allows us to view the above as a sequence of transformations of 2-dimensional string diagrams, which is built entirely out of snake introduction/cancellation moves and interchange, rendered in 3D as an animation: The swallowtail coherence asserts that this is equivalent to the ‘do-nothing’ identity transformation. Nick Hu202518https://forest.nickx.hu/coherent-inverses-AW7B/coherent-inverses-AW7B/coherent-inverses-AW7B/Swallowtail source 3D animationfigure The source of the swallowtail 4D diagram, rendered at time . In the first Morse projection, we construct this by contraction resulting in the following zigzag homotopy: Viewed as a 4-dimensional string diagram, as an animation of 3-dimensional surface diagrams, this zigzag homotopy looks like figure . As before, this is a different (one of many possible) swallowtail coherence to the one in chapter . Nick Hu202518https://forest.nickx.hu/coherent-inverses-AG5Q/coherent-inverses-AG5Q/coherent-inverses-AG5Q/Swallowtail coherence 4D animationfigure Swallowtail coherence, rendered at time .Nick Hu2023123https://forest.nickx.hu/coherent-inverses-applications-catastrophes-butterfly/coherent-inverses-applications-catastrophes-butterfly/coherent-inverses-applications-catastrophes-butterfly/Butterfly coherencesection A butterfly coherence is a 5-dimensional coherence on a swallowtail. In the first Morse projection, this is given by the assertion that the zigzag homotopy: is equivalent to the identity 4-cell. Equivalently, figure allows us to view the above as a sequence of transformations of 3-dimensional surface diagrams, which is built entirely out of swallowtail introduction/cancellation moves and interchange, rendered in 4D as an animation: The butterfly coherence asserts that this is equivalent to the ‘do-nothing’ identity transformation. Nick Hu202518https://forest.nickx.hu/coherent-inverses-Q0TG/coherent-inverses-Q0TG/coherent-inverses-Q0TG/Butterfly source 4D animationfigure The source of the butterfly 5D diagram, rendered at time . In the second Morse projection, we construct this by contraction resulting in the following zigzag homotopy: Link to online proof: https://beta.homotopy.io/p/2412.00002. Nick Hu2023123https://forest.nickx.hu/coherent-inverses-applications-figure-8-isotopy/coherent-inverses-applications-figure-8-isotopy/coherent-inverses-applications-figure-8-isotopy/Figure-8 isotopysection The homotopy hypothesis establishes a correspondence between topological spaces (up to homotopy equivalence) with -groupoids. Moreover, an -groupoid is merely a higher category for which every -cell, for 0]]>, is invertible. We can utilise this connection to present spaces algebraically with string diagrams, which we then manipulate within homotopy.io to deduce theorems about topology. A benefit of this approach is that the geometry of string diagrams aligns nicely with topological intuition, allowing for an algebraic formalisation of essentially topological arguments. The free -groupoid on a 0-cell and a single 2-cell generates the homotopy type of the 2-sphere , where the homotopy classes of the 3-cells in any given Hom set are in bijection with the elements of the homotopy group . Let us examine the Hom set . The identity 3-cell , corresponding to the unit element of , as a 2-dimensional string diagram in the first Morse projection is given by: There is a ‘figure-8’ 3-cell as follows, with its corresponding inverse: The witness for this invertibility, in dimension 4 (i.e. the invertible 4-cell ) is given by the zigzag homotopy: However, there is another ‘figure-8’ along with its inverse, obtained similarly to , but with the over/under braiding swapped: This pair of 3-cells is a priori different; however, using the coherences for the invertibility of , we can deduce that this is related by homotopy to the original pair. Nick Hu20241225https://forest.nickx.hu/coherent-inverses-0D6X/coherent-inverses-0D6X/coherent-inverses-0D6X/Figure-8 isotopytheorem With respect to , the 3-cells and are homotopic: Nick Hu20241225proof By the following zigzag homotopy: Nick Hu20241225https://forest.nickx.hu/coherent-inverses-NJGH/coherent-inverses-NJGH/coherent-inverses-NJGH/Third homotopy group of the 2-sphere is the integerscorollary. Nick Hu20241225proof In string diagrams, consider the set of generators of corresponding to some (composite) braiding, e.g. Such a generator can be decomposed into a composition of figure-8 3-cells by the invertibility of , in particular along the zigzag homotopy: E.g. Any such generator is therefore homotopic to some composition of , using the fact that there are a priori four possible figure-8 3-cells arranged into two pairs of inverses, both of which coincide (theorem ). Note that this set of generators extends to all of by homotopy along the snake homotopies admissible by the invertibility of , so we deduce that is freely generated by a single non-trivial generator so it is isomorphic to the integers under addition: . The string diagrammatic calculus for braided monoidal categories can be thought of as a 3-dimensional calculus of tubes, projected onto 2-dimensional space. Without projecting to 2D, homotopy.io can render the figure-8 3-cell in 3D as in figure . Nick Hu202518https://forest.nickx.hu/coherent-inverses-M0LN/coherent-inverses-M0LN/coherent-inverses-M0LN/Figure-8 3D tube diagramfigure Figure-8 3-cell in 3D, with occlusion for the braiding. The 4-cell isotopy that is then the proof of theorem can be rendered as an animation of tubes, as in figure . Nick Hu202518https://forest.nickx.hu/coherent-inverses-RP6J/coherent-inverses-RP6J/coherent-inverses-RP6J/Figure-8 isotopy 4D tube animationfigure Figure-8 isotopy, rendered at time . Link to online proof: https://beta.homotopy.io/p/2412.00001. Nick Hu https://forest.nickx.hu/double-categories-reading-group/ double-categories-reading-group /double-categories-reading-group/ Double Categories Reading Group Nick Hu 2025 6 3 Plan The plan is to meet every week or two to read seminal works on double categories and their applications to category theory and computer science. Currently, the most likely meeting time is Tuesday afternoon, and we are unlikely to start meeting until late July. I have in my mind some themes and papers which I would like to read, but I am pretty open to suggestions and redirection in general. In particular, I think it's best to follow a somewhat chronological order, to appreciate the wider context of why this technique was developed. I am hopeful that we can keep a communal ‘reading log’ on this page to keep track of the interesting things we learn and to add some accountability to our reading. The list (roughly ordered and sectioned): Introductory Higher Dimensional Categories: From Double to Multiple Categories — selected chapters From bicategories to (pseudo) double categories Framed bicategories and monoidal fibrations Cartesian Double Categories with an Emphasis on Characterizing Spans Constructing symmetric monoidal bicategories functorially Virtual equipments A unified framework for generalized multicategories String Diagrams For Double Categories and Equipments The formal theory of relative monads Nick Hu Nick Hu 2025 8 13 Higher Dimensional Categories: From Double to Multiple Categories — § 3.1 – § 3.4 Note Nick Hu Nick Hu 2025 8 13 Double category Definition A (strict) double category has a set of objects ; horizontal morphisms of the form (function-like), which compose by (often elided); vertical morphisms of the form (relation-like), which compose (diagrammatically) by ; double cells written inline as vertical and horizontal composition of double cells denoted by and respectively such that interchange (coherence) holds; i.e. the diagram is well-defined (and similarly identities cohere correctly). A double category is flat if its double cells are determined by their boundary horizontal/vertical morphisms. Nick Hu Nick Hu 2025 8 13 Example Let denote a relation from to , i.e. a subset of , and its relational converse: Moreover, by treating functions as relations which are everywhere uniquely defined on their domain, we can (pre/post) compose a relation by a function. Double cells are determined by where as above, with denoting subset inclusion of . This happens if and only if in . Note that this makes flat. This is an instance of the category of relations on a regular category (in this case, ). Nick Hu Nick Hu 2025 8 13 Quintets of 2-categories Example Let be a 2-category. The double category of quintets of , , has morphisms of as both horizontal and vertical morphisms, with its double cells derived from the 2-cells of . When is an ordinary category, viewed as a 2-category with trivial 2-cells, is the flat double category of commutative squares of . contains a cellwise full double subcategory of small categories with limit-preserving functors as horizontal morphisms and colimit-preserving functors as vertical morphisms. Nick Hu Nick Hu 2025 8 13 Double category of adjunctions Example Internal to any 2-category there is a notion of adjunction. Firstly, recall that an adjunction is given by a pair of antiparallel 1-morphisms equipped with unit-counit pair of 2-cells satisfying the triangle identities. That is, let denote such an antiparallel pair of 1-morphisms, with In , the triangle equations look like equalities of double cells: (there are symmetric versions of these diagrams too where the horizontal and vertical morphisms are swapped). Given a pair of morphisms and with another adjunction , we define the mate of any 2-cell to be pasting by and : and similarly for the mate of : These operations are mutually inverse by the triangle identities. This is a generalisation of the Hom set bijection generated by a pair of adjoint functors (e.g. currying). The double category of adjunctions, , has horizontal morphisms given by the 1-morphisms of and vertical morphisms given by adjunctions in . Double cells are given by mate pairs of 2-cells in . As a special case, is the double category of (small) categories, functors, and adjunctions. It is also equipped with a vertical involution given by taking opposite categories. Nick Hu Nick Hu 2025 8 13 Dualities Definition Unlike ordinary categories, double categories admit multiple distinct notions of duality: horizontal opposite , vertical opposite , and transpose (exchanging vertical and horizontal morphisms). The double dual (in any respect) is the identity; the horizontal-vertical dual is the vertical-horizontal dual; the horizontal-transpose dual is the transpose-vertical dual, and the transpose-horizontal dual is the vertical-transpose dual. 'co' generally refers to horizontal duality. Nick Hu Nick Hu 2025 8 13 Double categories as internal categories Definition In any category with pullbacks, there is a notion of internal category, presented by two objects and : is the object of objects, and is the object of morphisms; the above diagram stipulates that every object has an associated identity morphism, every morphism has source and target objects (moreover, is a common section of and ), and pairs of composable morphisms can be composed, subject to a bunch of commuting diagrams expressing the laws of composition. A (strict) double category (ignoring size issues) can be seen an internal category in in two ways. In the (main) horizontal way, is the category of objects and horizontal morphisms, is the category of vertical morphisms and double cells with horizontal composition, and is the category of pairs of composable vertical morphisms. The other (vertical) way is given by the horizontal interpretation with respect to the transpose of the double category. Nick Hu Nick Hu 2025 8 13 Double categories and 2-categories Definition A double category gives rise to two 2-categories: the horizontal 2-category where objects are those of , morphisms are the horizontal morphisms of , and (globular) 2-cells are given by double cells whose vertical morphisms are identities; the vertical 2-category where objects are those of , morphisms are the vertical morphisms of , and (special) 2-cells are given by double cells whose horizontal morphisms are identities. We also have that . Nick Hu Nick Hu 2025 8 13 Ultraflatness Definition A double category is called ultraflat if it is flat and both its horizontal and vertical 2-categories have thin 2-cell structure. That is, it is flat and any double cell which is horizontally or vertically invertible is an identity. Nick Hu Nick Hu 2025 8 13 Double functor double category Definition is the double category of double functors from to , given by objects double functors ; horizontal morphisms horizontal transformations of double functors , i.e. a double functor such that and ; explicitly, this is the data given by a horizontal morphism of for every object of ; a double cell of for every vertical morphism of , such that for every double cell of ; ; ; vertical morphisms vertical transformations of double functors , i.e. a double functor transversally to the above; explicitly: a vertical morphism of fo every object of ; a double cell of for every horizontal morphism of , such that for every double cell of ; ; ; double cells for two horizontal transformations and two vertical transformations , a modification which assigns to objects of double cells of such that for every horizontal morphism in : ; for every vertical morphism in : . Nick Hu Nick Hu 2025 8 13 Weak double category Definition A weak double category is a double category for which its vertical composition () has been weakened to being witnessed by special 2-cells which are invertible in the vertical 2-category. It can be viewed as a pseudo category object in . Horizontal composition is still strict. A strict double category is one where the comparison double cells are all identities. A weak double category is unitary when and are identities (strict units); it is preunitary when This condition is often required to remove pathologies. If is a weak double category, then the obvious notion of is a bicategory instead of a 2-category. If is preunitary, then is a well-defined 2-category. Nick Hu Nick Hu 2025 8 13 Example Let be a category with pullbacks. Spans in form a bicategory, where the composition of spans is given by pullback; it is weak because pullbacks are only determined up to unique isomorphism. is the amalgamation of this bicategory with to form a weak double category; it has objects objects of ; horizontal morphisms morphisms of ; vertical morphisms spans in , which can also be seen as diagrams where is the formal span category ; double cells natural transformations between spans, given as diagrams. This weak double category is unitary, and admits as and the bicategory of spans in as . There is a dual double category of cospans also. Nick Hu Nick Hu 2025 8 13 — the weak double category of profunctors Example A profunctor is defined as a functor . There is a bicategory of (small) categories, profunctors, and natural transformations (weak because composition of profunctors is given by coends). is the amalgamation of this with the category to form a weak double category ; it has objects (small) categories; horizontal morphisms functors; vertical morphisms profunctors; double cells for is given by a natural transformation The prototypical example of a profunctor is the Hom functor (this is also the identity profunctor). An arbitrary profunctor can be seen as a kind of 'heteromorphism' from the objects of to the objects of ; there is a category which is known as the gluing or collage of and along , whose objects and morphisms are the disjoint union of those coming from and , with new ones between given by the 'Hom set' for an object of and an object of (this is a double colimit known as the cotabulator of in ). This category should naturally be seen as fibred over , whose fibres over and are and respectively; the coend in the composition of profunctors captures an equivalence relation, for where and are 'heteromorphisms' and is a morphism in , generated by . Contributions