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Spectral sequences via linear presheaves 通过线性预波的谱序列
arXiv - MATH - Category Theory Pub Date : 2024-06-04 DOI: arxiv-2406.02777
Muriel Livernet, Sarah Whitehouse
{"title":"Spectral sequences via linear presheaves","authors":"Muriel Livernet, Sarah Whitehouse","doi":"arxiv-2406.02777","DOIUrl":"https://doi.org/arxiv-2406.02777","url":null,"abstract":"We study homotopy theory of the category of spectral sequences with respect\u0000to the class of weak equivalences given by maps which are quasi-isomorphisms on\u0000a fixed page. We introduce the category of extended spectral sequences and show\u0000that this is bicomplete by analysis of a certain linear presheaf category\u0000modelled on discs. We endow the category of extended spectral sequences with\u0000various model category structures, restricting to give the almost Brown\u0000category structures on spectral sequences of our earlier work. One of these has\u0000the property that spectral sequences is a homotopically full subcategory. By\u0000results of Meier, this exhibits the category of spectral sequences as a fibrant\u0000object in the Barwick-Kan model structure on relative categories, that is, it\u0000gives a model for an infinity category of spectral sequences. We also use the\u0000presheaf approach to define two d'ecalage functors on spectral sequences, left\u0000and right adjoint to a shift functor, thereby clarifying prior use of the term\u0000d'ecalage in connection with spectral sequences.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551891","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
A simple characterization of Quillen adjunctions 奎伦邻接的简单表征
arXiv - MATH - Category Theory Pub Date : 2024-06-04 DOI: arxiv-2406.02194
Victor Carmona
{"title":"A simple characterization of Quillen adjunctions","authors":"Victor Carmona","doi":"arxiv-2406.02194","DOIUrl":"https://doi.org/arxiv-2406.02194","url":null,"abstract":"We observe that an enriched right adjoint functor between model categories\u0000which preserves acyclic fibrations and fibrant objects is quite generically a\u0000right Quillen functor.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252373","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
A classifying localic category for locally compact locales with application to the Axiom of Infinity (poster) 局部紧凑局部的分类局部范畴及其在无穷公理中的应用（海报）
arXiv - MATH - Category Theory Pub Date : 2024-06-03 DOI: arxiv-2406.01573
Christopher Francis Townsend
{"title":"A classifying localic category for locally compact locales with application to the Axiom of Infinity (poster)","authors":"Christopher Francis Townsend","doi":"arxiv-2406.01573","DOIUrl":"https://doi.org/arxiv-2406.01573","url":null,"abstract":"For an internal category $mathbb{C}$ in a cartesian category $mathcal{C}$\u0000we define, naturally in objects $X$ of $mathcal{C}$, $Prin_{mathbb{C}}(X)$.\u0000This is a category whose objects are principal $c mathbb{C}$-bundles over $X$\u0000and whose morphisms are principal $c(mathbb{C}^{uparrow})$-bundles. Here\u0000$c(_)$ denotes taking the core groupoid of a category (same objects but only\u0000isomorphisms as morphisms) and $mathbb{C}^{uparrow}$ is the arrow category of\u0000$mathbb{C}$ (objects morphisms, morphisms commuting squares). We show that $X\u0000mapsto Prin_{mathbb{C}}(X)$ is a stack of categories and call stacks of this\u0000sort lax-geometric. We then provide two sufficient conditions for a stack to be\u0000lax-geometric and use them to prove that the pseudo-functor $X mapsto\u0000mathbf{LK}_{Sh(X)}$ on the category of locales $mathbf{Loc}$ is a\u0000lax-geometric stack. Here $mathbf{LK}_{Sh(X)}$ is the category of locally\u0000compact locales in the topos of sheaves over $X$, $Sh(X)$. Therefore there\u0000exists a localic category $mathbb{C}_{mathbf{LK}}$ such that\u0000$mathbf{LK}_{Sh(X)} simeq Prin_{mathbb{C}_{mathbf{LK}}}(X)$ naturally for\u0000every locale $X$. We then show how this can be used to give a new localic characterisation of\u0000the Axiom of Infinity.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253202","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
Strict refinement property of connected loop-free categories 连通无环范畴的严格细化属性
arXiv - MATH - Category Theory Pub Date : 2024-06-03 DOI: arxiv-2406.01106
Aly-Bora UlusoyCosynus, Emmanuel HaucourtCosynus
{"title":"Strict refinement property of connected loop-free categories","authors":"Aly-Bora UlusoyCosynus, Emmanuel HaucourtCosynus","doi":"arxiv-2406.01106","DOIUrl":"https://doi.org/arxiv-2406.01106","url":null,"abstract":"In this paper we study the strict refinement property for connected partial\u0000ordersalso known as Hashimoto's Theorem. This property implies that any\u0000isomorphismbetween products of irreducible structures is determined is uniquely\u0000determinedas a product of isomorphisms between the factors. This refinement\u0000implies asort of smallest possible decomposition for such structures. After a\u0000brief recallof the necessary notion we prove that Hashimoto's theorem can be\u0000extendedto connected loop-free categories, i.e. categories with no non-trivial\u0000morphismsendomorphisms. A special case of such categories is the category of\u0000connectedcomponents, for concurrent programs without loops.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252810","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
Groupoidal and truncated $n$-quasi-categories 类群和截断的 $n$ 准范畴
arXiv - MATH - Category Theory Pub Date : 2024-06-03 DOI: arxiv-2406.01490
Victor Brittes
{"title":"Groupoidal and truncated $n$-quasi-categories","authors":"Victor Brittes","doi":"arxiv-2406.01490","DOIUrl":"https://doi.org/arxiv-2406.01490","url":null,"abstract":"We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are\u0000the translation to the world of $n$-quasi-categories of groupoidal and\u0000truncated $(infty, n)$-$Theta$-spaces defined by Rezk. We show that these\u0000objects are the fibrant objects of model structures on the category of\u0000presheaves on $Theta_n$ obtained by localisation of Ara's model structure for\u0000$n$-quasi-categories. Furthermore, we prove that the inclusion $Delta to\u0000Theta_n$ induces a Quillen equivalence between the model structure for\u0000groupoidal (resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen\u0000model structure for spaces (resp. homotopy $n$-types) on simplicial sets. To\u0000get to these results, we also construct a cylinder object for\u0000$n$-quasi-categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252703","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
Construct ideal cotorsion pairs by recollement of triangulated categories 通过三角范畴的重归纳，构建理想的共轭对
arXiv - MATH - Category Theory Pub Date : 2024-06-01 DOI: arxiv-2406.00417
Qikai Wang, Haiyan Zhu
{"title":"Construct ideal cotorsion pairs by recollement of triangulated categories","authors":"Qikai Wang, Haiyan Zhu","doi":"arxiv-2406.00417","DOIUrl":"https://doi.org/arxiv-2406.00417","url":null,"abstract":"Let $(mathcal{T}',mathcal{T},mathcal{T}'')$ be a recollement of\u0000triangulated categories. Two complete ideal cotorsion pairs in $mathcal{T}'$\u0000and $mathcal{T}''$ can be induced by a complete ideal cotorsion pair in\u0000$mathcal{T}$. If $(mathcal{I},mathcal{I}^perp )$ and\u0000$(mathcal{J},mathcal{J}^perp)$ are two complete ideal cotorsion pair in\u0000triangulated category, then\u0000$(mathcal{I}capmathcal{J},langlemathcal{I}^perp,mathcal{J}^perprangle)$\u0000is also a complete ideal cotorsion pair. In this way, a series of ideal\u0000cotorsion pairs in $mathcal{T}$ can be induced by two ideal cotorsion pairs in\u0000$mathcal{T}'$ and $mathcal{T}''$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252620","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
Map monoidales and duoidal $infty$-categories 映射单元和二元$infty$类
arXiv - MATH - Category Theory Pub Date : 2024-05-31 DOI: arxiv-2406.00223
Takeshi Torii
{"title":"Map monoidales and duoidal $infty$-categories","authors":"Takeshi Torii","doi":"arxiv-2406.00223","DOIUrl":"https://doi.org/arxiv-2406.00223","url":null,"abstract":"In this paper we give an example of duoidal $infty$-categories. We introduce\u0000map $mathcal{O}$-monoidales in an $mathcal{O}$-monoidal $(infty,2)$-category\u0000for an $infty$-operad $mathcal{O}^{otimes}$. We show that the endomorphism\u0000mapping $infty$-category of a map $mathcal{O}$-monoidale is a coCartesian\u0000$(Delta^{rm op},mathcal{O})$-duoidal $infty$-category. After that, we\u0000introduce a convolution product on the mapping $infty$-category from an\u0000$mathcal{O}$-comonoidale to an $mathcal{O}$-monoidale. We show that the\u0000$mathcal{O}$-monoidal structure on the duoidal endomorphism mapping\u0000$infty$-category of a map $mathcal{O}$-monoidale is equivalent to the\u0000convolution product on the mapping $infty$-category from the dual\u0000$mathcal{O}$-comonoidale to the map $mathcal{O}$-monoidale.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253085","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
Profinite completions of products 产品的无限完备性
arXiv - MATH - Category Theory Pub Date : 2024-05-31 DOI: arxiv-2406.00136
Peter J. Haine
{"title":"Profinite completions of products","authors":"Peter J. Haine","doi":"arxiv-2406.00136","DOIUrl":"https://doi.org/arxiv-2406.00136","url":null,"abstract":"A source of difficulty in profinite homotopy theory is that the profinite\u0000completion functor does not preserve finite products. In this note, we provide\u0000a new, checkable criterion on prospaces $X$ and $Y$ that guarantees that the\u0000profinite completion of $Xtimes Y$ agrees with the product of the profinite\u0000completions of $X$ and $Y$. Using this criterion, we show that profinite\u0000completion preserves products of '{e}tale homotopy types of qcqs schemes. This\u0000fills a gap in Chough's proof of the K\"{u}nneth formula for the '{e}tale\u0000homotopy type of a product of proper schemes over a separably closed field.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252621","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
An Ultrametric for Cartesian Differential Categories for Taylor Series Convergence 泰勒级数收敛的笛卡尔微分类超计量
arXiv - MATH - Category Theory Pub Date : 2024-05-29 DOI: arxiv-2405.19474
Jean-Simon Pacaud Lemay
{"title":"An Ultrametric for Cartesian Differential Categories for Taylor Series Convergence","authors":"Jean-Simon Pacaud Lemay","doi":"arxiv-2405.19474","DOIUrl":"https://doi.org/arxiv-2405.19474","url":null,"abstract":"Cartesian differential categories provide a categorical framework for\u0000multivariable differential calculus and also the categorical semantics of the\u0000differential $lambda$-calculus. Taylor series expansion is an important\u0000concept for both differential calculus and the differential $lambda$-calculus.\u0000In differential calculus, a function is equal to its Taylor series if its\u0000sequence of Taylor polynomials converges to the function in the analytic sense.\u0000On the other hand, for the differential $lambda$-calculus, one works in a\u0000setting with an appropriate notion of algebraic infinite sums to formalize\u0000Taylor series expansion. In this paper, we provide a formal theory of Taylor\u0000series in an arbitrary Cartesian differential category without the need for\u0000converging limits or infinite sums. We begin by developing the notion of Taylor\u0000polynomials of maps in a Cartesian differential category and then show how\u0000comparing Taylor polynomials of maps induces an ultrapseudometric on the\u0000homsets. We say that a Cartesian differential category is Taylor if maps are\u0000entirely determined by their Taylor polynomials. The main results of this paper\u0000are that in a Taylor Cartesian differential category, the induced\u0000ultrapseudometrics are ultrametrics and that for every map $f$, its Taylor\u0000series converges to $f$ with respect to this ultrametric. This framework\u0000recaptures both Taylor series expansion in differential calculus via analytic\u0000methods and in categorical models of the differential $lambda$-calculus (or\u0000Differential Linear Logic) via infinite sums.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189274","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
Bi-directional models of `radically synthetic' differential geometry 彻底合成 "微分几何的双向模型
arXiv - MATH - Category Theory Pub Date : 2024-05-28 DOI: arxiv-2405.17748
Matías Menni
{"title":"Bi-directional models of `radically synthetic' differential geometry","authors":"Matías Menni","doi":"arxiv-2405.17748","DOIUrl":"https://doi.org/arxiv-2405.17748","url":null,"abstract":"The radically synthetic foundation for smooth geometry formulated in [Law11]\u0000postulates a space T with the property that it has a unique point and, out of\u0000the monoid T^T of endomorphisms, it extracts a submonoid R which, in many\u0000cases, is the (commutative) multiplication of a rig structure. The rig R is\u0000said to be bi-directional if its subobject of invertible elements has two\u0000connected components. In this case, R may be equipped with a pre-order\u0000compatible with the rig structure. We adjust the construction of `well-adapted'\u0000models of Synthetic Differential Geometry in order to build the first\u0000pre-cohesive toposes with a bi-directional R. We also show that, in one of\u0000these pre-cohesive variants, the pre-order on R, derived radically\u0000synthetically from bi-directionality, coincides with that defined in the\u0000original model.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166779","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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