Working within enriched category theory, we further develop the use of soundness, introduced by Ad'amek, Borceux, Lack, and Rosick'y for ordinary categories. In particular we investigate: (1) the theory of locally $Phi$-presentable $mathcal V$-categories for a sound class $Phi$, (2) the problem of whether every $Phi$-accessible $mathcal V$-category is $Psi$-accessible, for given sound classes $PhisubseteqPsi$, and (3) a notion of $Phi$-ary equational theory whose $mathcal V$-categories of models characterize algebras for $Phi$-ary monads on $mathcal V$.
{"title":"More on soundness in the enriched context","authors":"Giacomo Tendas","doi":"arxiv-2409.00389","DOIUrl":"https://doi.org/arxiv-2409.00389","url":null,"abstract":"Working within enriched category theory, we further develop the use of\u0000soundness, introduced by Ad'amek, Borceux, Lack, and Rosick'y for ordinary\u0000categories. In particular we investigate: (1) the theory of locally\u0000$Phi$-presentable $mathcal V$-categories for a sound class $Phi$, (2) the\u0000problem of whether every $Phi$-accessible $mathcal V$-category is\u0000$Psi$-accessible, for given sound classes $PhisubseteqPsi$, and (3) a\u0000notion of $Phi$-ary equational theory whose $mathcal V$-categories of models\u0000characterize algebras for $Phi$-ary monads on $mathcal V$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225576","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}
Danel Ahman, Greta Coraglia, Davide Castelnovo, Fosco Loregian, Nelson Martins-Ferreira, Ülo Reimaa
We study fibrations arising from indexed categories of the following form: fix two categories $mathcal{A},mathcal{X}$ and a functor $F : mathcal{A} times mathcal{X} longrightarrowmathcal{X} $, so that to each $F_A=F(A,-)$ one can associate a category of algebras $mathbf{Alg}_mathcal{X}(F_A)$ (or an Eilenberg-Moore, or a Kleisli category if each $F_A$ is a monad). We call the functor $int^{mathcal{A}}mathbf{Alg} to mathcal{A}$, whose typical fibre over $A$ is the category $mathbf{Alg}_mathcal{X}(F_A)$, the "fibration of algebras" obtained from $F$. Examples of such constructions arise in disparate areas of mathematics, and are unified by the intuition that $int^mathcal{A}mathbf{Alg} $ is a form of semidirect product of the category $mathcal{A}$, acting on $mathcal{X}$, via the `representation' given by the functor $F : mathcal{A} times mathcal{X} longrightarrowmathcal{X}$. After presenting a range of examples and motivating said intuition, the present work focuses on comparing a generic fibration with a fibration of algebras: we prove that if $mathcal{A}$ has an initial object, under very mild assumptions on a fibration $p : mathcal{E}longrightarrow mathcal{A}$, we can define a canonical action of $mathcal{A}$ letting it act on the fibre $mathcal{E}_varnothing$ over the initial object. This result bears some resemblance to the well-known fact that the fundamental group $pi_1(B)$ of a base space acts naturally on the fibers $F_b = p^{-1}b$ of a fibration $p : E to B$.
{"title":"Fibrations of algebras","authors":"Danel Ahman, Greta Coraglia, Davide Castelnovo, Fosco Loregian, Nelson Martins-Ferreira, Ülo Reimaa","doi":"arxiv-2408.16581","DOIUrl":"https://doi.org/arxiv-2408.16581","url":null,"abstract":"We study fibrations arising from indexed categories of the following form:\u0000fix two categories $mathcal{A},mathcal{X}$ and a functor $F : mathcal{A}\u0000times mathcal{X} longrightarrowmathcal{X} $, so that to each $F_A=F(A,-)$\u0000one can associate a category of algebras $mathbf{Alg}_mathcal{X}(F_A)$ (or an\u0000Eilenberg-Moore, or a Kleisli category if each $F_A$ is a monad). We call the\u0000functor $int^{mathcal{A}}mathbf{Alg} to mathcal{A}$, whose typical fibre\u0000over $A$ is the category $mathbf{Alg}_mathcal{X}(F_A)$, the \"fibration of\u0000algebras\" obtained from $F$. Examples of such constructions arise in disparate areas of mathematics, and\u0000are unified by the intuition that $int^mathcal{A}mathbf{Alg} $ is a form of\u0000semidirect product of the category $mathcal{A}$, acting on $mathcal{X}$, via\u0000the `representation' given by the functor $F : mathcal{A} times mathcal{X}\u0000longrightarrowmathcal{X}$. After presenting a range of examples and motivating said intuition, the\u0000present work focuses on comparing a generic fibration with a fibration of\u0000algebras: we prove that if $mathcal{A}$ has an initial object, under very mild\u0000assumptions on a fibration $p : mathcal{E}longrightarrow mathcal{A}$, we can\u0000define a canonical action of $mathcal{A}$ letting it act on the fibre\u0000$mathcal{E}_varnothing$ over the initial object. This result bears some\u0000resemblance to the well-known fact that the fundamental group $pi_1(B)$ of a\u0000base space acts naturally on the fibers $F_b = p^{-1}b$ of a fibration $p : E\u0000to B$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199175","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 show that any functor between $infty$-categories can be straightened. More precisely, we show that for any $infty$-category $mathcal{C}$, there is an equivalence between the $infty$-category $(mathrm{Cat}_{infty})_{/mathcal{C}}$ of $infty$-categories over $mathcal{C}$ and the $infty$-category of unital lax functors from $mathcal{C}$ to the double $infty$-category $mathrm{Corr}$ of correspondences. The proof relies on a certain universal property of the Morita category which is of independent interest.
{"title":"On the straightening of every functor","authors":"Thomas Blom","doi":"arxiv-2408.16539","DOIUrl":"https://doi.org/arxiv-2408.16539","url":null,"abstract":"We show that any functor between $infty$-categories can be straightened.\u0000More precisely, we show that for any $infty$-category $mathcal{C}$, there is\u0000an equivalence between the $infty$-category\u0000$(mathrm{Cat}_{infty})_{/mathcal{C}}$ of $infty$-categories over\u0000$mathcal{C}$ and the $infty$-category of unital lax functors from\u0000$mathcal{C}$ to the double $infty$-category $mathrm{Corr}$ of\u0000correspondences. The proof relies on a certain universal property of the Morita\u0000category which is of independent interest.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199191","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 establish a bi-equivalence between the bi-category of topoi with enough points and a localisation of a bi-subcategory of topological groupoids
我们在有足够点的拓扑的双类别和拓扑群的双子类的局部化之间建立了双等价关系
{"title":"Topoi with enough points and topological groupoids","authors":"Joshua Wrigley","doi":"arxiv-2408.15848","DOIUrl":"https://doi.org/arxiv-2408.15848","url":null,"abstract":"We establish a bi-equivalence between the bi-category of topoi with enough\u0000points and a localisation of a bi-subcategory of topological groupoids","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199176","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 a novel model that interprets the parallel operator, also present in algebraic calculi, within the context of propositional logic. This interpretation uses the category $mathbf{Mag}_{mathbf{Set}}$, whose objects are magmas and whose arrows are functions from the category $mathbf{Set}$, specifically for the case of the parallel lambda calculus. Similarly, we use the category $mathbf{AMag}^{mathcal S}_{mathbf{Set}}$, whose objects are action magmas and whose arrows are also functions from the category $mathbf{Set}$, for the case of the algebraic lambda calculus. Our approach diverges from conventional interpretations where disjunctions are handled by coproducts, instead proposing to handle them with the union of disjoint union and the Cartesian product.
{"title":"Parallel and algebraic lambda-calculi in intuitionistic propositional logic","authors":"Alejandro Díaz-Caro, Octavio Malherbe","doi":"arxiv-2408.16102","DOIUrl":"https://doi.org/arxiv-2408.16102","url":null,"abstract":"We introduce a novel model that interprets the parallel operator, also\u0000present in algebraic calculi, within the context of propositional logic. This\u0000interpretation uses the category $mathbf{Mag}_{mathbf{Set}}$, whose objects\u0000are magmas and whose arrows are functions from the category $mathbf{Set}$,\u0000specifically for the case of the parallel lambda calculus. Similarly, we use\u0000the category $mathbf{AMag}^{mathcal S}_{mathbf{Set}}$, whose objects are\u0000action magmas and whose arrows are also functions from the category\u0000$mathbf{Set}$, for the case of the algebraic lambda calculus. Our approach\u0000diverges from conventional interpretations where disjunctions are handled by\u0000coproducts, instead proposing to handle them with the union of disjoint union\u0000and the Cartesian product.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225577","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 study the framework of $infty$-equipments which is designed to produce well-behaved theories for different generalizations of $infty$-categories in a synthetic and uniform fashion. We consider notions of (lax) functors between these equipments, closed monoidal structures on these equipments, and fibrations internal to these equipments. As a main application, we will demonstrate that the foundations of internal $infty$-category theory can be readily obtained using this formalism.
{"title":"Formal category theory in $infty$-equipments II: Lax functors, monoidality and fibrations","authors":"Jaco Ruit","doi":"arxiv-2408.15190","DOIUrl":"https://doi.org/arxiv-2408.15190","url":null,"abstract":"We study the framework of $infty$-equipments which is designed to produce\u0000well-behaved theories for different generalizations of $infty$-categories in a\u0000synthetic and uniform fashion. We consider notions of (lax) functors between\u0000these equipments, closed monoidal structures on these equipments, and\u0000fibrations internal to these equipments. As a main application, we will\u0000demonstrate that the foundations of internal $infty$-category theory can be\u0000readily obtained using this formalism.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199177","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 this paper, we study double $infty$-categories of double functors. To this end, we exhibit the cartesian closed structure of the $infty$-category of double $infty$-categories and various localizations. We prove a theorem that characterizes the companions and conjoints in functor double $infty$-categories via the notion of companionable and conjointable 2-cells in double $infty$-categories. Moreover, we show that under suitable conditions, functor double $infty$-categories are horizontally closed. Throughout the paper, we highlight a few applications to $(infty,2)$-category theory and indexed exponentiability.
{"title":"On functor double $infty$-categories","authors":"Jaco Ruit","doi":"arxiv-2408.14335","DOIUrl":"https://doi.org/arxiv-2408.14335","url":null,"abstract":"In this paper, we study double $infty$-categories of double functors. To\u0000this end, we exhibit the cartesian closed structure of the $infty$-category of\u0000double $infty$-categories and various localizations. We prove a theorem that\u0000characterizes the companions and conjoints in functor double\u0000$infty$-categories via the notion of companionable and conjointable 2-cells in\u0000double $infty$-categories. Moreover, we show that under suitable conditions,\u0000functor double $infty$-categories are horizontally closed. Throughout the\u0000paper, we highlight a few applications to $(infty,2)$-category theory and\u0000indexed exponentiability.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225578","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 investigate the most general gauging operations in 2+1 dimensional oriented field theories with finite symmetry groups, which correspond to gapped boundary conditions in 3+1 dimensional Dijkgraaf-Witten theory. The classification is achieved by enumerating 2+1 dimensional oriented topological quantum field theories that cancel the 't Hooft anomaly associated with the symmetry. This framework is rigorously formulated using twisted crossed extensions of modular fusion categories and projective 3-representations. Additionally, we explore the resulting fusion 2-category symmetries and argue that this framework captures all possible categorical symmetries in 2+1 dimensional oriented field theories.
{"title":"Towards All Categorical Symmetries in 2+1 Dimensions","authors":"Mathew Bullimore, Jamie J. Pearson","doi":"arxiv-2408.13931","DOIUrl":"https://doi.org/arxiv-2408.13931","url":null,"abstract":"We investigate the most general gauging operations in 2+1 dimensional\u0000oriented field theories with finite symmetry groups, which correspond to gapped\u0000boundary conditions in 3+1 dimensional Dijkgraaf-Witten theory. The\u0000classification is achieved by enumerating 2+1 dimensional oriented topological\u0000quantum field theories that cancel the 't Hooft anomaly associated with the\u0000symmetry. This framework is rigorously formulated using twisted crossed\u0000extensions of modular fusion categories and projective 3-representations.\u0000Additionally, we explore the resulting fusion 2-category symmetries and argue\u0000that this framework captures all possible categorical symmetries in 2+1\u0000dimensional oriented field theories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199178","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}
Extriangulated categories, introduced by Nakaoka and Palu, serve as a simultaneous generalization of exact and triangulated categories. In this paper, we first introduce the concept of admissible weak factorization systems and establish a bijection between cotorsion pairs and admissible weak factorization systems in extriangulated categories. Consequently, we give the equivalences between hereditary cotorsion pairs and compatible cotorsion pairs via admissible weak factorization systems under certain conditions in extriangulated categories, thereby generalizing a result by Di, Li, and Liang.
{"title":"Admissible weak factorization systems on extriangulated categories","authors":"Yajun Ma, Hanyang You, Dongdong Zhang, Panyue Zhou","doi":"arxiv-2408.13548","DOIUrl":"https://doi.org/arxiv-2408.13548","url":null,"abstract":"Extriangulated categories, introduced by Nakaoka and Palu, serve as a\u0000simultaneous generalization of exact and triangulated categories. In this\u0000paper, we first introduce the concept of admissible weak factorization systems\u0000and establish a bijection between cotorsion pairs and admissible weak\u0000factorization systems in extriangulated categories. Consequently, we give the\u0000equivalences between hereditary cotorsion pairs and compatible cotorsion pairs\u0000via admissible weak factorization systems under certain conditions in\u0000extriangulated categories, thereby generalizing a result by Di, Li, and Liang.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225579","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}
Lakshya Bhardwaj, Thibault Décoppet, Sakura Schafer-Nameki, Matthew Yu
We study the fusion 3-categorical symmetries for quantum theories in (3+1)d with self-duality defects. Such defects have been realized physically by half-space gauging in theories with 1-form symmetries $A[1]$ for an abelian group $A$, and have found applications in the continuum and the lattice. These fusion 3-categories will be called (generalized) Tambara-Yamagami fusion 3-categories $(mathbf{3TY})$. We consider the Brauer-Picard and Picard 4-groupoids to construct these categories using a 3-categorical version of the extension theory introduced by Etingof, Nikshych and Ostrik. These two 4-groupoids correspond to the construction of duality defects either directly in 4d, or from the 5d Symmetry Topological Field Theory (SymTFT). The Witt group of non-degenerate braided fusion 1-categories naturally appears in the aforementioned 4-groupoids and represents enrichments of standard duality defects by (2+1)d TFTs. Our main objective is to study graded extensions of the fusion 3-category $mathbf{3Vect}(A[1])$. Firstly, we use invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of $mathbf{3Vect}(A[1])$ can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of $mathbf{3Vect}(A[1])$, which represents topological defects of the SymTFT, is completely described by a sylleptic strongly fusion 2-category formed by topological surface defects of the SymTFT. These are classified by a finite abelian group equipped with an alternating 2-form. We relate the Picard 4-groupoid of the corresponding braided fusion 3-categories with a generalized Witt group constructed from certain graded braided fusion 1-categories using a twisted Deligne tensor product. We perform explicit computations for $mathbb{Z}/2$ and $mathbb{Z}/4$ graded $mathbf{3TY}$ categories.
{"title":"Fusion 3-Categories for Duality Defects","authors":"Lakshya Bhardwaj, Thibault Décoppet, Sakura Schafer-Nameki, Matthew Yu","doi":"arxiv-2408.13302","DOIUrl":"https://doi.org/arxiv-2408.13302","url":null,"abstract":"We study the fusion 3-categorical symmetries for quantum theories in (3+1)d\u0000with self-duality defects. Such defects have been realized physically by\u0000half-space gauging in theories with 1-form symmetries $A[1]$ for an abelian\u0000group $A$, and have found applications in the continuum and the lattice. These\u0000fusion 3-categories will be called (generalized) Tambara-Yamagami fusion\u00003-categories $(mathbf{3TY})$. We consider the Brauer-Picard and Picard\u00004-groupoids to construct these categories using a 3-categorical version of the\u0000extension theory introduced by Etingof, Nikshych and Ostrik. These two\u00004-groupoids correspond to the construction of duality defects either directly\u0000in 4d, or from the 5d Symmetry Topological Field Theory (SymTFT). The Witt\u0000group of non-degenerate braided fusion 1-categories naturally appears in the\u0000aforementioned 4-groupoids and represents enrichments of standard duality\u0000defects by (2+1)d TFTs. Our main objective is to study graded extensions of the\u0000fusion 3-category $mathbf{3Vect}(A[1])$. Firstly, we use invertible bimodule\u00003-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the\u0000Brauer-Picard 4-groupoid of $mathbf{3Vect}(A[1])$ can be identified with the\u0000Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of\u0000$mathbf{3Vect}(A[1])$, which represents topological defects of the SymTFT, is\u0000completely described by a sylleptic strongly fusion 2-category formed by\u0000topological surface defects of the SymTFT. These are classified by a finite\u0000abelian group equipped with an alternating 2-form. We relate the Picard\u00004-groupoid of the corresponding braided fusion 3-categories with a generalized\u0000Witt group constructed from certain graded braided fusion 1-categories using a\u0000twisted Deligne tensor product. We perform explicit computations for\u0000$mathbb{Z}/2$ and $mathbb{Z}/4$ graded $mathbf{3TY}$ categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225580","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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