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The Homotopy Theory of $A_infty$Categories A_infty$类的同调理论
arXiv - MATH - Category Theory Pub Date : 2024-08-09 DOI: arxiv-2408.05325
Mattia Ornaghi
{"title":"The Homotopy Theory of $A_infty$Categories","authors":"Mattia Ornaghi","doi":"arxiv-2408.05325","DOIUrl":"https://doi.org/arxiv-2408.05325","url":null,"abstract":"In this paper we describe the homotopy category of the $A_infty$categories.\u0000To do that we introduce the notion of semi-free $A_infty$category, which plays\u0000the role of standard cofibration. Moreover, we define the non unital $A_infty$\u0000(resp. DG)categories with cofibrant morphisms and we prove that any non unital\u0000$A_infty$ (resp. DG)category has a resolution of this kind.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225581","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}
引用次数: 0
Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I 具有非不可逆对称性的 (2+1)d 中的空隙相位：第一部分
arXiv - MATH - Category Theory Pub Date : 2024-08-09 DOI: arxiv-2408.05266
Lakshya Bhardwaj, Daniel Pajer, Sakura Schafer-Nameki, Apoorv Tiwari, Alison Warman, Jingxiang Wu
{"title":"Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I","authors":"Lakshya Bhardwaj, Daniel Pajer, Sakura Schafer-Nameki, Apoorv Tiwari, Alison Warman, Jingxiang Wu","doi":"arxiv-2408.05266","DOIUrl":"https://doi.org/arxiv-2408.05266","url":null,"abstract":"We use the Symmetry Topological Field Theory (SymTFT) to study and classify\u0000gapped phases in (2+1)d for a class of categorical symmetries, referred to as\u0000being of bosonic type. The SymTFTs for these symmetries are given by twisted\u0000and untwisted (3+1)d Dijkgraaf-Witten (DW) theories for finite groups G. A\u0000finite set of boundary conditions (BCs) of these DW theories is well-known:\u0000these simply involve imposing Dirichlet and Neumann conditions on the (3+1)d\u0000gauge fields. We refer to these as minimal BCs. The key new observation here is\u0000that for each DW theory, there exists an infinite number of other BCs, that we\u0000call non-minimal BCs. These non-minimal BCs are all obtained by a 'theta\u0000construction', which involves stacking the Dirichlet BC with 3d TFTs having G\u00000-form symmetry, and gauging the diagonal G symmetry. On the one hand, using\u0000the non-minimal BCs as symmetry BCs gives rise to an infinite number of\u0000non-invertible symmetries having the same SymTFT, while on the other hand,\u0000using the non-minimal BCs as physical BCs in the sandwich construction gives\u0000rise to an infinite number of (2+1)d gapped phases for each such non-invertible\u0000symmetry. Our analysis is thoroughly exemplified for G = $mathbb{Z_2}$ and\u0000more generally any finite abelian group, for which the resulting non-invertible\u0000symmetries and their gapped phases already reveal an immensely rich structure.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199188","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}
引用次数: 0
$(infty,n)$-Limits II: Comparison across models $(infty,n)$-限制 II：不同模型之间的比较
arXiv - MATH - Category Theory Pub Date : 2024-08-08 DOI: arxiv-2408.04742
Lyne Moser, Martina Rovelli, Nima Rasekh
{"title":"$(infty,n)$-Limits II: Comparison across models","authors":"Lyne Moser, Martina Rovelli, Nima Rasekh","doi":"arxiv-2408.04742","DOIUrl":"https://doi.org/arxiv-2408.04742","url":null,"abstract":"We show that the notion of $(infty,n)$-limit defined using the enriched\u0000approach and the one defined using the internal approach coincide. We also give\u0000explicit constructions of various double $(infty,n-1)$-categories implementing\u0000various join constructions, slice constructions and cone constructions, and\u0000study their properties. We further prove that key examples of\u0000$(infty,n)$-categories are (co)complete.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938699","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}
引用次数: 0
When do CF-approximation spaces capture sL-domains CF 近似空间何时捕获 sL 域
arXiv - MATH - Category Theory Pub Date : 2024-08-07 DOI: arxiv-2408.03529
Guojun WuNanjing University of Information Science and Technology, Luoshan XuYangzhou University, Wei YaoNanjing University of Information Science and Technology
{"title":"When do CF-approximation spaces capture sL-domains","authors":"Guojun WuNanjing University of Information Science and Technology, Luoshan XuYangzhou University, Wei YaoNanjing University of Information Science and Technology","doi":"arxiv-2408.03529","DOIUrl":"https://doi.org/arxiv-2408.03529","url":null,"abstract":"In this paper, by means of upper approximation operators in rough set theory,\u0000we study representations for sL-domains and its special subclasses. We\u0000introduce the concepts of sL-approximation spaces, L-approximation spaces and\u0000bc-approximation spaces, which are special types of CF-approximation spaces. We\u0000prove that the collection of CF-closed sets in an sL-approximation space\u0000(resp., an L-approximation space, a bc-approximation space) ordered by\u0000set-theoretic inclusion is an sL-domain (resp., an L-domain, a bc-domain);\u0000conversely, every sL-domain (resp., L-domain, bc-domain) is order-isomorphic to\u0000the collection of CF-closed sets of an sL-approximation space (resp., an\u0000L-approximation space, a bc-approximation space). Consequently, we establish an\u0000equivalence between the category of sL-domains (resp., L-domains) with Scott\u0000continuous mappings and that of sL-approximation spaces (resp., L-approximation\u0000spaces) with CF-approximable relations.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938786","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}
引用次数: 0
Representations of FS-domains and BF-domains via FS-approximation Spaces 通过 FS-approximation Spaces 表示 FS 域和 BF 域
arXiv - MATH - Category Theory Pub Date : 2024-08-07 DOI: arxiv-2408.03523
Guojun WuNanjing University of Information Science and Technology, Luoshan XuYangzhou University
{"title":"Representations of FS-domains and BF-domains via FS-approximation Spaces","authors":"Guojun WuNanjing University of Information Science and Technology, Luoshan XuYangzhou University","doi":"arxiv-2408.03523","DOIUrl":"https://doi.org/arxiv-2408.03523","url":null,"abstract":"In this paper, concepts of (topological) FS-approximation spaces are\u0000introduced. Representations of FS-domains and BF-domains via (topological)\u0000FS-approximation spaces are considered. It is proved that the collection of\u0000CF-closed sets in an FS-approximation space (resp., a topological\u0000FS-approximation space) endowed with the set-inclusion order is an FS-domain\u0000(resp., a BF-domain) and that every FS-domain (resp., BF-domain) is order\u0000isomorphic to the collection of CF-closed sets of some FS-approximation space\u0000(resp., topological FS-approximation space) endowed with the set-inclusion\u0000order. The concept of topological BF-approximation spaces is introduced and a\u0000skillful method without using CF-approximable relations to represent BF-domains\u0000is given. It is also proved that the category of FS-domains (resp., BF-domains)\u0000with Scott continuous maps as morphisms is equivalent to that of\u0000FS-approximation spaces (resp., topological FS-approximation spaces) with\u0000CF-approximable relations as morphisms.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938783","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}
引用次数: 0
Hearts of set-generated t-structures have a set of generators 心集生成的 t 结构有一组生成器
arXiv - MATH - Category Theory Pub Date : 2024-08-02 DOI: arxiv-2408.01378
Manuel Saorín
{"title":"Hearts of set-generated t-structures have a set of generators","authors":"Manuel Saorín","doi":"arxiv-2408.01378","DOIUrl":"https://doi.org/arxiv-2408.01378","url":null,"abstract":"We show that if $alpha$ is a regular cardinal, $mathcal{D}$ is an\u0000$alpha$-compactly generated triangulated category, in the sense of Neeman\u0000cite{N}, and $tau$ is a t-structure in $mathcal{D}$ generated by a set of\u0000$alpha$-compact objects, then the heart of $tau$ is a locally\u0000$alpha$-presentable (not necessarily Ab5) abelian category. As a consequence,\u0000in a well-generated triangulated category any t-structure generated by a set of\u0000objects has a heart with a set of generators.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938782","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}
引用次数: 0
Gray (skew) multicategories: double and Gray-categorical cases 灰色（倾斜）多类别：双重和灰色类别情况
arXiv - MATH - Category Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00561
Bojana Femić
{"title":"Gray (skew) multicategories: double and Gray-categorical cases","authors":"Bojana Femić","doi":"arxiv-2408.00561","DOIUrl":"https://doi.org/arxiv-2408.00561","url":null,"abstract":"We construct in a unifying way skew-multicategories and multicategories of\u0000double and Gray-categories that we call Gray (skew) multicategories. We study\u0000their different versions depending on the types of functors and higher\u0000transforms. We construct Gray type products by generators and relations and\u0000prove that Gray skew-multicategories are closed and representable on one side,\u0000and that the Gray multicaticategories taken with the strict type of functors\u0000are representable. We conclude that the categories of double and\u0000Gray-categories with strict functors underlying Gray (skew) multicategories are\u0000skew monoidal, respectively monoidal, depending on the type of the inner-hom\u0000and product considered. The described Gray (skew) multicategories we see as\u0000prototypes of general Gray (skew) multicategories, which correspond to (higher)\u0000categories of higher dimensional internal and enriched categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881568","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}
引用次数: 0
Quantale-valued maps and partial maps 量值映射和局部映射
arXiv - MATH - Category Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00393
Lili Shen, Xiaoye Tang
{"title":"Quantale-valued maps and partial maps","authors":"Lili Shen, Xiaoye Tang","doi":"arxiv-2408.00393","DOIUrl":"https://doi.org/arxiv-2408.00393","url":null,"abstract":"Let $mathsf{Q}$ be a commutative and unital quantale. By a $mathsf{Q}$-map\u0000we mean a left adjoint in the quantaloid of sets and $mathsf{Q}$-relations,\u0000and by a partial $mathsf{Q}$-map we refer to a Kleisli morphism with respect\u0000to the maybe monad on the category $mathsf{Q}text{-}mathbf{Map}$ of sets and\u0000$mathsf{Q}$-maps. It is shown that every $mathsf{Q}$-map is symmetric if and\u0000only if $mathsf{Q}$ is weakly lean, and that every $mathsf{Q}$-map is exactly\u0000a map in $mathbf{Set}$ if and only $mathsf{Q}$ is lean. Moreover, assuming\u0000the axiom of choice, it is shown that the category of sets and partial\u0000$mathsf{Q}$-maps is monadic over $mathsf{Q}text{-}mathbf{Map}$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141887406","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}
引用次数: 0
The least subtopos containing the discrete skeleton of $Ω$ 包含 $Ω$ 离散骨架的最小子顶
arXiv - MATH - Category Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00514
Matí as Menni
{"title":"The least subtopos containing the discrete skeleton of $Ω$","authors":"Matí as Menni","doi":"arxiv-2408.00514","DOIUrl":"https://doi.org/arxiv-2408.00514","url":null,"abstract":"Let $p: mathcal{E} to mathcal{S}$ be a pre-cohesive geometric morphism. We\u0000show that the least subtopos of $mathcal{E}$ containing both the subcategories\u0000$p^*: mathcal{S} to mathcal{E}$ and $p^!: mathcal{S} to mathcal{E}$\u0000exists, and that it coincides with the least subtopos containing $p^*2$, where\u00002 denotes the subobject classifier of $mathcal{S}$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881567","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}
引用次数: 0
Coproduct idempotent algebras over internal operads in enriched $infty$-categories 丰富$infty$类别中内部操作数上的共品幂等价代数
arXiv - MATH - Category Theory Pub Date : 2024-07-31 DOI: arxiv-2407.21706
Federico Ernesto Mocchetti
{"title":"Coproduct idempotent algebras over internal operads in enriched $infty$-categories","authors":"Federico Ernesto Mocchetti","doi":"arxiv-2407.21706","DOIUrl":"https://doi.org/arxiv-2407.21706","url":null,"abstract":"In arXiv:1712.00555, H. Heine shows that given a symmetric monoidal\u0000$infty$-category $mathcal{V}$ and a weakly $mathcal{V}$-enriched monad $T$\u0000over an $infty$-category $mathcal{C}$, then there is an induced action of\u0000$mathcal{V}$ on $LMod_T(mathcal{C})$. Moreover, properties like tensoring or\u0000enrichment can be transferred from the action on $mathcal{C}$ to that on\u0000$LMod_T(mathcal{C})$. We see that the action of an internal operad $O in\u0000Alg(sSeq(mathcal{C}))$ can be interpreted as the action of a monad $T_O$, such\u0000that $Alg_O(mathcal{C})cong LMod_{T_O}(mathcal{C})$. We can then prove that,\u0000under a presentability assumption, if the category $mathcal{C}$ admits\u0000cotensors with respect to the action of $mathcal{V}$, then so does\u0000$Alg_O(mathcal{C})cong LMod_{T_O}(mathcal{C})$. This is used to show that\u0000the coproduct-idempotent algebras are fixed by the induced tensoring action. We\u0000apply this to the stable motivic homotopy category and prove that the tensor of\u0000any motivic sphere with rational motivic cohomology is equivalent to the\u0000latter.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869296","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}
引用次数: 0
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