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Applied Categorical Structures最新文献

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Koszul Monoids in Quasi-abelian Categories 准阿贝尔范畴中的科斯祖尔单体
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-12-06 DOI: 10.1007/s10485-023-09756-7
Rhiannon Savage
{"title":"Koszul Monoids in Quasi-abelian Categories","authors":"Rhiannon Savage","doi":"10.1007/s10485-023-09756-7","DOIUrl":"10.1007/s10485-023-09756-7","url":null,"abstract":"<div><p>Suppose that we have a bicomplete closed symmetric monoidal quasi-abelian category <span>(mathcal {E})</span> with enough flat projectives, such as the category of complete bornological spaces <span>({{textbf {CBorn}}}_k)</span> or the category of inductive limits of Banach spaces <span>({{textbf {IndBan}}}_k)</span>. Working with monoids in <span>(mathcal {E})</span>, we can generalise and extend the Koszul duality theory of Beilinson, Ginzburg, Soergel. We use an element-free approach to define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders’ embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of certain subcategories of the derived categories of graded modules over Koszul monoids and their duals.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09756-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homotopy Sheaves on Generalised Spaces 广义空间上的同伦轴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-12-04 DOI: 10.1007/s10485-023-09754-9
Severin Bunk
{"title":"Homotopy Sheaves on Generalised Spaces","authors":"Severin Bunk","doi":"10.1007/s10485-023-09754-9","DOIUrl":"10.1007/s10485-023-09754-9","url":null,"abstract":"<div><p>We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding preserves homotopy sheaves valued in (sufficiently nice) simplicial model categories. Moreover, we show that this induces an equivalence between sheaves of spaces on the original category and colimit-preserving sheaves of spaces on its free cocompletion. We present three applications in geometry and topology: first, we prove that diffeological vector bundles descend along subductions of diffeological spaces. Second, we deduce that various flavours of bundle gerbes with connection satisfy <span>((infty ,2))</span>-categorical descent. Finally, we investigate smooth diffeomorphism actions in smooth bordism-type field theories on a manifold. We show how these smooth actions allow us to extract the values of a field theory on any object coherently from its values on generating objects of the bordism category.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09754-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138485110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Unbounded Algebraic Derivators 无界代数导数
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-12-02 DOI: 10.1007/s10485-023-09752-x
Leovigildo Alonso Tarrío, Beatriz Álvarez Díaz, Ana Jeremías López
{"title":"Unbounded Algebraic Derivators","authors":"Leovigildo Alonso Tarrío,&nbsp;Beatriz Álvarez Díaz,&nbsp;Ana Jeremías López","doi":"10.1007/s10485-023-09752-x","DOIUrl":"10.1007/s10485-023-09752-x","url":null,"abstract":"<div><p>We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138475583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories 一般组合模型范畴中经由同伦末端的同伦极限
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-11-27 DOI: 10.1007/s10485-023-09747-8
Sergey Arkhipov, Sebastian Ørsted
{"title":"Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories","authors":"Sergey Arkhipov,&nbsp;Sebastian Ørsted","doi":"10.1007/s10485-023-09747-8","DOIUrl":"10.1007/s10485-023-09747-8","url":null,"abstract":"<div><p>We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield–Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138449161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Partialising Institutions Partialising机构
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-11-15 DOI: 10.1007/s10485-023-09753-w
Răzvan Diaconescu
{"title":"Partialising Institutions","authors":"Răzvan Diaconescu","doi":"10.1007/s10485-023-09753-w","DOIUrl":"10.1007/s10485-023-09753-w","url":null,"abstract":"<div><p><span>({3/2})</span>-Institutions have been introduced as an extension of institution theory that accommodates implicitly partiality of the signature morphisms together with its syntactic and semantic effects. In this paper we show that ordinary institutions that are equipped with an inclusion system for their categories of signatures generate naturally <span>({3/2})</span>-institutions with <i>explicit</i> partiality for their signature morphisms. This provides a general uniform way to build <span>({3/2})</span>-institutions for the foundations of conceptual blending and software evolution. Moreover our general construction allows for an uniform derivation of some useful technical properties.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134796639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Compact Hausdorff Locales in Presheaf Toposes Presheaf拓扑中的紧凑Hausdorff区域
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-10-19 DOI: 10.1007/s10485-023-09751-y
Simon Henry, Christopher Townsend
{"title":"Compact Hausdorff Locales in Presheaf Toposes","authors":"Simon Henry,&nbsp;Christopher Townsend","doi":"10.1007/s10485-023-09751-y","DOIUrl":"10.1007/s10485-023-09751-y","url":null,"abstract":"<div><p>We prove that for any small category <span>({mathcal {C}})</span>, the category <span>(textbf{KHausLoc}_{hat{{mathcal {C}}}})</span> of compact Hausdorff locales in the presheaf topos <span>(hat{{mathcal {C}}})</span>, is equivalent to the category of functors <span>({mathcal {C}} rightarrow textbf{KHausLoc})</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49999601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Continuous Nakayama Representations 连续中山表示
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-10-03 DOI: 10.1007/s10485-023-09748-7
Job Daisie Rock, Shijie Zhu
{"title":"Continuous Nakayama Representations","authors":"Job Daisie Rock,&nbsp;Shijie Zhu","doi":"10.1007/s10485-023-09748-7","DOIUrl":"10.1007/s10485-023-09748-7","url":null,"abstract":"<div><p>We introduce continuous analogues of Nakayama algebras. In particular, we introduce the notion of (pre-)Kupisch functions, which play a role as Kupisch series of Nakayama algebras, and view a continuous Nakayama representation as a special type of representation of <span>({mathbb {R}})</span> or <span>({mathbb {S}}^1)</span>. We investigate equivalences and connectedness of the categories of Nakayama representations. Specifically, we prove that orientation-preserving homeomorphisms on <span>({mathbb {R}})</span> and on <span>({mathbb {S}}^1)</span> induce equivalences between these categories. Connectedness is characterized by a special type of points called separation points determined by (pre-)Kupisch functions. We also construct an exact embedding from the category of finite-dimensional representations for any finite-dimensional Nakayama algebra, to a category of continuous Nakayama representaitons.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50007434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Pervin Spaces and Frith Frames: Bitopological Aspects and Completion Pervin空间与Frith框架：双拓扑方面与完成
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-09-30 DOI: 10.1007/s10485-023-09749-6
Célia Borlido, Anna Laura Suarez
{"title":"Pervin Spaces and Frith Frames: Bitopological Aspects and Completion","authors":"Célia Borlido,&nbsp;Anna Laura Suarez","doi":"10.1007/s10485-023-09749-6","DOIUrl":"10.1007/s10485-023-09749-6","url":null,"abstract":"<div><p>A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of <span>(T_0)</span> complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr’s characterizations of sober and <span>(T_D)</span> topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09749-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50055551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness 从gs -一元到Oplax笛卡尔范畴:构造与功能完备性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-09-28 DOI: 10.1007/s10485-023-09750-z
Tobias Fritz, Fabio Gadducci, Davide Trotta, Andrea Corradini
{"title":"From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness","authors":"Tobias Fritz,&nbsp;Fabio Gadducci,&nbsp;Davide Trotta,&nbsp;Andrea Corradini","doi":"10.1007/s10485-023-09750-z","DOIUrl":"10.1007/s10485-023-09750-z","url":null,"abstract":"<div><p>Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to <span>(textbf{Rel})</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50052357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Nonexistence of Colimits in Naive Discrete Homotopy Theory Naive离散同调理论中Colimits的不存在性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-09-27 DOI: 10.1007/s10485-023-09746-9
Daniel Carranza, Krzysztof Kapulkin, Jinho Kim
{"title":"Nonexistence of Colimits in Naive Discrete Homotopy Theory","authors":"Daniel Carranza,&nbsp;Krzysztof Kapulkin,&nbsp;Jinho Kim","doi":"10.1007/s10485-023-09746-9","DOIUrl":"10.1007/s10485-023-09746-9","url":null,"abstract":"<div><p>We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit colimits. In particular, we settle in the negative the question of whether the A-homotopy equivalences in the category of graphs are part of a model structure.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50103432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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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