In this paper we describe the homotopy category of the $A_infty$categories. To do that we introduce the notion of semi-free $A_infty$category, which plays the role of standard cofibration. Moreover, we define the non unital $A_infty$ (resp. DG)categories with cofibrant morphisms and we prove that any non unital $A_infty$ (resp. DG)category has a resolution of this kind.
{"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}
We use the Symmetry Topological Field Theory (SymTFT) to study and classify gapped phases in (2+1)d for a class of categorical symmetries, referred to as being of bosonic type. The SymTFTs for these symmetries are given by twisted and untwisted (3+1)d Dijkgraaf-Witten (DW) theories for finite groups G. A finite set of boundary conditions (BCs) of these DW theories is well-known: these simply involve imposing Dirichlet and Neumann conditions on the (3+1)d gauge fields. We refer to these as minimal BCs. The key new observation here is that for each DW theory, there exists an infinite number of other BCs, that we call non-minimal BCs. These non-minimal BCs are all obtained by a 'theta construction', which involves stacking the Dirichlet BC with 3d TFTs having G 0-form symmetry, and gauging the diagonal G symmetry. On the one hand, using the non-minimal BCs as symmetry BCs gives rise to an infinite number of non-invertible symmetries having the same SymTFT, while on the other hand, using the non-minimal BCs as physical BCs in the sandwich construction gives rise to an infinite number of (2+1)d gapped phases for each such non-invertible symmetry. Our analysis is thoroughly exemplified for G = $mathbb{Z_2}$ and more generally any finite abelian group, for which the resulting non-invertible symmetries and their gapped phases already reveal an immensely rich structure.
我们使用对称拓扑场理论（SymTFT）来研究和分类一类分类对称（被称为玻色类型）的(2+1)d中的隙相。这些对称性的 SymTFTs 是由有限群 G 的扭曲和非扭曲 (3+1)d Dijkgraaf-Witten (DW) 理论给出的。这些 DW 理论的边界条件（BCs）是众所周知的：这些条件只涉及对 (3+1)dge 场施加 Dirichlet 和 Neumann 条件。我们把它们称为最小边界条件。这里的关键新发现是，对于每一个DW理论，都存在着无限多的其他BC，我们称之为非最小BC。这些非最小 BC 都是通过 "thetaconstruction "得到的，其中包括用具有 G0 形式对称性的 3d TFT 堆叠 Dirichlet BC，并对对角线 G 对称性进行测量。一方面，使用非最小 BC 作为对称 BC 会产生无数个具有相同 SymTFT 的非不可逆对称；另一方面，在三明治结构中使用非最小 BC 作为物理 BC 会为每个非不可逆对称产生无数个 (2+1)d 间隙相。我们的分析对 G = $mathbb{Z_2}$ 以及更广义的任何有限无性群都做了详尽的举例说明，由此产生的非不对称及其间隙相已经揭示了极其丰富的结构。
{"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}
We show that the notion of $(infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(infty,n-1)$-categories implementing various join constructions, slice constructions and cone constructions, and study their properties. We further prove that key examples of $(infty,n)$-categories are (co)complete.
{"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}
Guojun WuNanjing University of Information Science and Technology, Luoshan XuYangzhou University, Wei YaoNanjing University of Information Science and Technology
In this paper, by means of upper approximation operators in rough set theory, we study representations for sL-domains and its special subclasses. We introduce the concepts of sL-approximation spaces, L-approximation spaces and bc-approximation spaces, which are special types of CF-approximation spaces. We prove that the collection of CF-closed sets in an sL-approximation space (resp., an L-approximation space, a bc-approximation space) ordered by set-theoretic inclusion is an sL-domain (resp., an L-domain, a bc-domain); conversely, every sL-domain (resp., L-domain, bc-domain) is order-isomorphic to the collection of CF-closed sets of an sL-approximation space (resp., an L-approximation space, a bc-approximation space). Consequently, we establish an equivalence between the category of sL-domains (resp., L-domains) with Scott continuous mappings and that of sL-approximation spaces (resp., L-approximation spaces) with CF-approximable relations.
{"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}
Guojun WuNanjing University of Information Science and Technology, Luoshan XuYangzhou University
In this paper, concepts of (topological) FS-approximation spaces are introduced. Representations of FS-domains and BF-domains via (topological) FS-approximation spaces are considered. It is proved that the collection of CF-closed sets in an FS-approximation space (resp., a topological FS-approximation space) endowed with the set-inclusion order is an FS-domain (resp., a BF-domain) and that every FS-domain (resp., BF-domain) is order isomorphic to the collection of CF-closed sets of some FS-approximation space (resp., topological FS-approximation space) endowed with the set-inclusion order. The concept of topological BF-approximation spaces is introduced and a skillful method without using CF-approximable relations to represent BF-domains is given. It is also proved that the category of FS-domains (resp., BF-domains) with Scott continuous maps as morphisms is equivalent to that of FS-approximation spaces (resp., topological FS-approximation spaces) with CF-approximable relations as morphisms.
{"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}
We show that if $alpha$ is a regular cardinal, $mathcal{D}$ is an $alpha$-compactly generated triangulated category, in the sense of Neeman cite{N}, and $tau$ is a t-structure in $mathcal{D}$ generated by a set of $alpha$-compact objects, then the heart of $tau$ is a locally $alpha$-presentable (not necessarily Ab5) abelian category. As a consequence, in a well-generated triangulated category any t-structure generated by a set of objects has a heart with a set of generators.
我们证明，如果$alpha$是一个正则红心，$mathcal{D}$是一个在Neemancite{N}意义上$alpha$紧凑生成的三角范畴，并且$tau$是$mathcal{D}$中由一组$alpha$紧凑对象生成的t结构，那么$tau$的心就是一个局部$alpha$可呈现（不一定是Ab5）的阿贝尔范畴。因此，在一个生成良好的三角范畴中，任何由一组对象生成的 t 结构都有一个具有一组生成子的心。
{"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}
We construct in a unifying way skew-multicategories and multicategories of double and Gray-categories that we call Gray (skew) multicategories. We study their different versions depending on the types of functors and higher transforms. We construct Gray type products by generators and relations and prove that Gray skew-multicategories are closed and representable on one side, and that the Gray multicaticategories taken with the strict type of functors are representable. We conclude that the categories of double and Gray-categories with strict functors underlying Gray (skew) multicategories are skew monoidal, respectively monoidal, depending on the type of the inner-hom and product considered. The described Gray (skew) multicategories we see as prototypes of general Gray (skew) multicategories, which correspond to (higher) categories of higher dimensional internal and enriched categories.
{"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}
Let $mathsf{Q}$ be a commutative and unital quantale. By a $mathsf{Q}$-map we mean a left adjoint in the quantaloid of sets and $mathsf{Q}$-relations, and by a partial $mathsf{Q}$-map we refer to a Kleisli morphism with respect to the maybe monad on the category $mathsf{Q}text{-}mathbf{Map}$ of sets and $mathsf{Q}$-maps. It is shown that every $mathsf{Q}$-map is symmetric if and only if $mathsf{Q}$ is weakly lean, and that every $mathsf{Q}$-map is exactly a map in $mathbf{Set}$ if and only $mathsf{Q}$ is lean. Moreover, assuming the axiom of choice, it is shown that the category of sets and partial $mathsf{Q}$-maps is monadic over $mathsf{Q}text{-}mathbf{Map}$.
{"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}
Let $p: mathcal{E} to mathcal{S}$ be a pre-cohesive geometric morphism. We show that the least subtopos of $mathcal{E}$ containing both the subcategories $p^*: mathcal{S} to mathcal{E}$ and $p^!: mathcal{S} to mathcal{E}$ exists, and that it coincides with the least subtopos containing $p^*2$, where 2 denotes the subobject classifier of $mathcal{S}$.
{"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}
In arXiv:1712.00555, H. Heine shows that given a symmetric monoidal $infty$-category $mathcal{V}$ and a weakly $mathcal{V}$-enriched monad $T$ over an $infty$-category $mathcal{C}$, then there is an induced action of $mathcal{V}$ on $LMod_T(mathcal{C})$. Moreover, properties like tensoring or enrichment can be transferred from the action on $mathcal{C}$ to that on $LMod_T(mathcal{C})$. We see that the action of an internal operad $O in Alg(sSeq(mathcal{C}))$ can be interpreted as the action of a monad $T_O$, such that $Alg_O(mathcal{C})cong LMod_{T_O}(mathcal{C})$. We can then prove that, under a presentability assumption, if the category $mathcal{C}$ admits cotensors with respect to the action of $mathcal{V}$, then so does $Alg_O(mathcal{C})cong LMod_{T_O}(mathcal{C})$. This is used to show that the coproduct-idempotent algebras are fixed by the induced tensoring action. We apply this to the stable motivic homotopy category and prove that the tensor of any motivic sphere with rational motivic cohomology is equivalent to the latter.
{"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}
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