In this survey article, we review some conceptual approaches to the cyclic category $Lambda$, as well as its description as a crossed simplicial group. We then give a new proof of the model structure on cyclic sets, work through the details of the generalized Reedy structure on cyclic spaces, and introduce model structures for cyclic Segal spaces and cyclic 2-Segal spaces.
{"title":"Cyclic Segal Spaces","authors":"Julia E. Bergner, Walker H. Stern","doi":"arxiv-2409.11945","DOIUrl":"https://doi.org/arxiv-2409.11945","url":null,"abstract":"In this survey article, we review some conceptual approaches to the cyclic\u0000category $Lambda$, as well as its description as a crossed simplicial group.\u0000We then give a new proof of the model structure on cyclic sets, work through\u0000the details of the generalized Reedy structure on cyclic spaces, and introduce\u0000model structures for cyclic Segal spaces and cyclic 2-Segal spaces.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an unbiased theory of symmetric multicategories, where sequences are replaced by families. To be effective, this approach requires an explicit consideration of indexing and reindexing of objects and arrows, handled by the double category $dPb$ of pullback squares in finite sets: a symmetric multicategory is a sum preserving discrete fibration of double categories $M: dMto dPb$. If the "loose" part of $M$ is an opfibration we get unbiased symmetric monoidal categories. The definition can be usefully generalized by replacing $dPb$ with another double prop $dP$, as an indexing base, giving $dP$-multicategories. For instance, we can remove the finiteness condition to obtain infinitary symmetric multicategories, or enhance $dPb$ by totally ordering the fibers of its loose arrows to obtain plain multicategories. We show how several concepts and properties find a natural setting in this framework. We also consider cartesian multicategories as algebras for a monad $(-)^cart$ on $sMlt$, where the loose arrows of $dM^cart$ are "spans" of a tight and a loose arrow in $dM$.
{"title":"Unbiased multicategory theory","authors":"Claudio Pisani","doi":"arxiv-2409.10150","DOIUrl":"https://doi.org/arxiv-2409.10150","url":null,"abstract":"We present an unbiased theory of symmetric multicategories, where sequences\u0000are replaced by families. To be effective, this approach requires an explicit\u0000consideration of indexing and reindexing of objects and arrows, handled by the\u0000double category $dPb$ of pullback squares in finite sets: a symmetric\u0000multicategory is a sum preserving discrete fibration of double categories $M:\u0000dMto dPb$. If the \"loose\" part of $M$ is an opfibration we get unbiased\u0000symmetric monoidal categories. The definition can be usefully generalized by replacing $dPb$ with another\u0000double prop $dP$, as an indexing base, giving $dP$-multicategories. For\u0000instance, we can remove the finiteness condition to obtain infinitary symmetric\u0000multicategories, or enhance $dPb$ by totally ordering the fibers of its loose\u0000arrows to obtain plain multicategories. We show how several concepts and properties find a natural setting in this\u0000framework. We also consider cartesian multicategories as algebras for a monad\u0000$(-)^cart$ on $sMlt$, where the loose arrows of $dM^cart$ are \"spans\" of a\u0000tight and a loose arrow in $dM$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"190 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Partial difference operators for a large class of functors between presheaf categories are introduced, extending our difference operator from cite{Par24} to the multivariable case. These combine into the Jacobian profunctor which provides the setting for a lax chain rule. We introduce a functorial version of multivariable Newton series whose aim is to recover a functor from its iterated differences. Not all functors are recovered but we get a best approximation in the form of a left adjoint, and the induced comonad is idempotent. Its fixed points are what we call soft analytic functors, a generalization of the multivariable analytic functors of Fiore et al.~cite{FioGamHylWin08}.
{"title":"Multivariate functorial difference","authors":"Robert Paré","doi":"arxiv-2409.09494","DOIUrl":"https://doi.org/arxiv-2409.09494","url":null,"abstract":"Partial difference operators for a large class of functors between presheaf\u0000categories are introduced, extending our difference operator from cite{Par24}\u0000to the multivariable case. These combine into the Jacobian profunctor which\u0000provides the setting for a lax chain rule. We introduce a functorial version of\u0000multivariable Newton series whose aim is to recover a functor from its iterated\u0000differences. Not all functors are recovered but we get a best approximation in\u0000the form of a left adjoint, and the induced comonad is idempotent. Its fixed\u0000points are what we call soft analytic functors, a generalization of the\u0000multivariable analytic functors of Fiore et al.~cite{FioGamHylWin08}.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"157 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a local-global principle for $infty$-categories over any base $infty$-category $mathcal{C}$: we show that any $infty$-category $mathcal{B} to mathcal{C}$ over $mathcal{C}$ is determined by the following data: the collection of fibers $mathcal{B}_X$ for $X$ running through the set of equivalence classes of objects of $mathcal{C}$ endowed with the action of the space of automorphisms $mathrm{Aut}_X(mathcal{B})$ on the fiber, the local data, together with a locally cartesian fibration $mathcal{D} to mathcal{C}$ and $mathrm{Aut}_X(mathcal{B})$-linear equivalences $mathcal{D}_X simeq mathcal{P}(mathcal{B}_X)$ to the $infty$-category of presheaves on $mathcal{B}_X$, the gluing data. As applications we describe the $infty$-category of small $infty$-categories over $[1]$ in terms of the $infty$-category of left fibrations and prove an end formula for mapping spaces of the internal hom of the $infty$-category of small $infty$-categories over $[1]$ and the conditionally existing internal hom of the $infty$-category of small $infty$-categories over any small $infty$-category $mathcal{C}.$ Considering functoriality in $mathcal{C}$ we obtain as a corollary that the double $infty$-category $mathrm{CORR}$ of correspondences is the pullback of the double $infty$-category $mathrm{PR}^L$ of presentable $infty$-categories along the functor $inftymathrm{Cat} to mathrm{Pr}^L$ taking presheaves. We deduce that $infty$-categories over any $infty$-category $mathcal{C}$ are classified by normal lax 2-functors.
{"title":"A local-global principle for parametrized $infty$-categories","authors":"Hadrian Heine","doi":"arxiv-2409.05568","DOIUrl":"https://doi.org/arxiv-2409.05568","url":null,"abstract":"We prove a local-global principle for $infty$-categories over any base\u0000$infty$-category $mathcal{C}$: we show that any $infty$-category\u0000$mathcal{B} to mathcal{C}$ over $mathcal{C}$ is determined by the following\u0000data: the collection of fibers $mathcal{B}_X$ for $X$ running through the set\u0000of equivalence classes of objects of $mathcal{C}$ endowed with the action of\u0000the space of automorphisms $mathrm{Aut}_X(mathcal{B})$ on the fiber, the\u0000local data, together with a locally cartesian fibration $mathcal{D} to\u0000mathcal{C}$ and $mathrm{Aut}_X(mathcal{B})$-linear equivalences\u0000$mathcal{D}_X simeq mathcal{P}(mathcal{B}_X)$ to the $infty$-category of\u0000presheaves on $mathcal{B}_X$, the gluing data. As applications we describe the\u0000$infty$-category of small $infty$-categories over $[1]$ in terms of the\u0000$infty$-category of left fibrations and prove an end formula for mapping\u0000spaces of the internal hom of the $infty$-category of small\u0000$infty$-categories over $[1]$ and the conditionally existing internal hom of\u0000the $infty$-category of small $infty$-categories over any small\u0000$infty$-category $mathcal{C}.$ Considering functoriality in $mathcal{C}$ we\u0000obtain as a corollary that the double $infty$-category $mathrm{CORR}$ of\u0000correspondences is the pullback of the double $infty$-category $mathrm{PR}^L$\u0000of presentable $infty$-categories along the functor $inftymathrm{Cat} to\u0000mathrm{Pr}^L$ taking presheaves. We deduce that $infty$-categories over any\u0000$infty$-category $mathcal{C}$ are classified by normal lax 2-functors.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matteo Capucci, Geoffrey S. H. Cruttwell, Neil Ghani, Fabio Zanasi
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation, including cartesian differential categories, generalised cartesian differential categories, tangent categories, as well as the versions of these categories axiomatising reverse derivatives. We explain uniformly and concisely the requirements expressed by these structures, using sections of suitable fibrations as unifying concept. Our perspective sheds light on their similarities and differences, as well as simplifying certain constructions from the literature.
{"title":"A Fibrational Theory of First Order Differential Structures","authors":"Matteo Capucci, Geoffrey S. H. Cruttwell, Neil Ghani, Fabio Zanasi","doi":"arxiv-2409.05763","DOIUrl":"https://doi.org/arxiv-2409.05763","url":null,"abstract":"We develop a categorical framework for reasoning about abstract properties of\u0000differentiation, based on the theory of fibrations. Our work encompasses the\u0000first-order fragments of several existing categorical structures for\u0000differentiation, including cartesian differential categories, generalised\u0000cartesian differential categories, tangent categories, as well as the versions\u0000of these categories axiomatising reverse derivatives. We explain uniformly and\u0000concisely the requirements expressed by these structures, using sections of\u0000suitable fibrations as unifying concept. Our perspective sheds light on their\u0000similarities and differences, as well as simplifying certain constructions from\u0000the literature.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cross-connections of normal categories was introduced by K.S.S.Nambooripad while discussing the structure of regular semigroups and via this cross-connections he obtained a beautiful representetion of regualr semigroup called the cross-connection semigroup (see cf.[4]). Subsequently cross-connection representation of various other semigroups such as concordant semigroups, semigroup of endomorphisms of a vector space are also described (cf.[6][5]). In this paper we describe the semigroup amalgam of cross-connection semigroups of the fibers of a vector bundle.
{"title":"Cross-connection semigroups amalgam of a vector bundle","authors":"P. G. Romeo","doi":"arxiv-2409.05062","DOIUrl":"https://doi.org/arxiv-2409.05062","url":null,"abstract":"Cross-connections of normal categories was introduced by K.S.S.Nambooripad\u0000while discussing the structure of regular semigroups and via this\u0000cross-connections he obtained a beautiful representetion of regualr semigroup\u0000called the cross-connection semigroup (see cf.[4]). Subsequently\u0000cross-connection representation of various other semigroups such as concordant\u0000semigroups, semigroup of endomorphisms of a vector space are also described\u0000(cf.[6][5]). In this paper we describe the semigroup amalgam of\u0000cross-connection semigroups of the fibers of a vector bundle.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce presheaf automata as a generalisation of different variants of higher-dimensional automata and other automata-like formalisms, including Petri nets and vector addition systems. We develop the foundations of a language theory for them based on notions of paths and track objects. We also define open maps for presheaf automata, extending the standard notions of simulation and bisimulation for transition systems. Apart from these conceptual contributions, we show that certain finite-type presheaf automata subsume all Petri nets, generalising a previous result by van Glabbeek, which applies to higher-dimensional automata and safe Petri nets.
我们介绍的预设自动机是对高维自动机和其他类似自动机形式的不同变体（包括 Petrinets 和向量加法系统）的概括。我们以路径和轨迹对象的概念为基础，为它们建立了语言理论的基础。我们还定义了预叶自动机的开放映射，扩展了过渡系统的标准模拟和双模拟概念。除了这些概念上的贡献之外，我们还证明了某些有限型预设自动机包含所有 Petri 网，从而推广了 van Glabbeek 以前的一个结果，该结果适用于高维自动机和安全 Petri 网。
{"title":"Presheaf automata","authors":"Georg Struth, Krzysztof Ziemiański","doi":"arxiv-2409.04612","DOIUrl":"https://doi.org/arxiv-2409.04612","url":null,"abstract":"We introduce presheaf automata as a generalisation of different variants of\u0000higher-dimensional automata and other automata-like formalisms, including Petri\u0000nets and vector addition systems. We develop the foundations of a language\u0000theory for them based on notions of paths and track objects. We also define\u0000open maps for presheaf automata, extending the standard notions of simulation\u0000and bisimulation for transition systems. Apart from these conceptual\u0000contributions, we show that certain finite-type presheaf automata subsume all\u0000Petri nets, generalising a previous result by van Glabbeek, which applies to\u0000higher-dimensional automata and safe Petri nets.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In (Borceux-Janelidze 2001) they prove a Categorical Galois Theorem for ordinary categories, and establish the main result of (Joyal-Tierney 1984), along with the classical Galois theory of Rings, as instances of this more general result. The main result of the present work refines this to a Quasicategorical Galois Theorem, by drawing heavily on the foundation laid in (Lurie 2024). More importantly, the argument used to prove the result is intended to highlight a deep connection between factorization systems (specifically the lex modalities of (Anel-Biedermann-Finster-Joyal 2021)), higher-categorical Galois Theorems, and Galois theories internal to higher toposes. This is the first part in a series of works, intended merely to motivate the lens and prove Theorem 3.4. In future work, we will delve into a generalization of the argument, and offer tools for producing applications.
{"title":"The Unreasonable Efficacy of the Lifting Condition in Higher Categorical Galois Theory I: a Quasi-categorical Galois Theorem","authors":"Joseph Rennie","doi":"arxiv-2409.03347","DOIUrl":"https://doi.org/arxiv-2409.03347","url":null,"abstract":"In (Borceux-Janelidze 2001) they prove a Categorical Galois Theorem for\u0000ordinary categories, and establish the main result of (Joyal-Tierney 1984),\u0000along with the classical Galois theory of Rings, as instances of this more\u0000general result. The main result of the present work refines this to a\u0000Quasicategorical Galois Theorem, by drawing heavily on the foundation laid in\u0000(Lurie 2024). More importantly, the argument used to prove the result is\u0000intended to highlight a deep connection between factorization systems\u0000(specifically the lex modalities of (Anel-Biedermann-Finster-Joyal 2021)),\u0000higher-categorical Galois Theorems, and Galois theories internal to higher\u0000toposes. This is the first part in a series of works, intended merely to\u0000motivate the lens and prove Theorem 3.4. In future work, we will delve into a\u0000generalization of the argument, and offer tools for producing applications.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop the theory of tricategorical limits and colimits, and show that they can be modelled up to biequivalence via certain homotopically well-behaved limits and colimits enriched over the monoidal model category $mathbf{Gray}$ of $2$-categories and $2$-functors. This categorifies the relationship that bicategorical limits and colimits have with the so called `flexible' enriched limits in $2$-category theory. As examples, we establish the tricategorical universal properties of Kleisli constructions for pseudomonads, Eilenberg-Moore and Kleisli constructions for (op)monoidal pseudomonads, centre constructions for $mathbf{Gray}$-monoids, and strictifications of bicategories and pseudo-double categories.
{"title":"Tricategorical Universal Properties Via Enriched Homotopy Theory","authors":"Adrian Miranda","doi":"arxiv-2409.01837","DOIUrl":"https://doi.org/arxiv-2409.01837","url":null,"abstract":"We develop the theory of tricategorical limits and colimits, and show that\u0000they can be modelled up to biequivalence via certain homotopically well-behaved\u0000limits and colimits enriched over the monoidal model category $mathbf{Gray}$\u0000of $2$-categories and $2$-functors. This categorifies the relationship that\u0000bicategorical limits and colimits have with the so called `flexible' enriched\u0000limits in $2$-category theory. As examples, we establish the tricategorical\u0000universal properties of Kleisli constructions for pseudomonads, Eilenberg-Moore\u0000and Kleisli constructions for (op)monoidal pseudomonads, centre constructions\u0000for $mathbf{Gray}$-monoids, and strictifications of bicategories and\u0000pseudo-double categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We initiate the combinatorial study of the poset $mathrm{wIndex}_{mathcal{T}}$ of weak $mathcal{T}$-indexing systems, consisting of composable collections of arities for $mathcal{T}$-equivariant algebraic structures, where $mathcal{T}$ is an orbital $infty$-category, such as the orbit category of a finite group. In particular, we show that these are equivalent to weak $mathcal{T}$-indexing categories and characterize various unitality conditions. Within this sits a natural generalization $mathrm{Index}_{mathcal{T}} subset mathrm{wIndex}_{mathcal{T}}$ of Blumberg-Hill's indexing systems, consisting of arities for structures possessing binary operations and unit elements. We characterize the relationship between the posets of unital weak indexing systems and indexing systems, the latter remaining isomorphic to transfer systems on this level of generality. We use this to characterize the poset of unital $C_{p^n}$-weak indexing systems.
{"title":"Orbital categories and weak indexing systems","authors":"Natalie Stewart","doi":"arxiv-2409.01377","DOIUrl":"https://doi.org/arxiv-2409.01377","url":null,"abstract":"We initiate the combinatorial study of the poset\u0000$mathrm{wIndex}_{mathcal{T}}$ of weak $mathcal{T}$-indexing systems,\u0000consisting of composable collections of arities for $mathcal{T}$-equivariant\u0000algebraic structures, where $mathcal{T}$ is an orbital $infty$-category, such\u0000as the orbit category of a finite group. In particular, we show that these are\u0000equivalent to weak $mathcal{T}$-indexing categories and characterize various\u0000unitality conditions. Within this sits a natural generalization $mathrm{Index}_{mathcal{T}}\u0000subset mathrm{wIndex}_{mathcal{T}}$ of Blumberg-Hill's indexing systems,\u0000consisting of arities for structures possessing binary operations and unit\u0000elements. We characterize the relationship between the posets of unital weak\u0000indexing systems and indexing systems, the latter remaining isomorphic to\u0000transfer systems on this level of generality. We use this to characterize the\u0000poset of unital $C_{p^n}$-weak indexing systems.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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