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Applied Categorical Structures最新文献

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Codescent and Bicolimits of Pseudo-Algebras 伪代数的编码和双极限
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-04-09 DOI: 10.1007/s10485-024-09765-0
Axel Osmond
{"title":"Codescent and Bicolimits of Pseudo-Algebras","authors":"Axel Osmond","doi":"10.1007/s10485-024-09765-0","DOIUrl":"10.1007/s10485-024-09765-0","url":null,"abstract":"<div><p>We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of codescent objects. After prerequisites on pseudomonads and their pseudo-algebras, we give a 2-dimensional Linton theorem reducing bicocompleteness of 2-categories of pseudo-algebras to existence of bicoequalizers of codescent objects. Finally we prove this condition to be fulfilled in the case of a bifinitary pseudomonad, ensuring bicocompleteness.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568364","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
Morita Equivalence and Morita Duality for Rings with Local Units and the Subcategory of Projective Unitary Modules 带局部单元的环的莫里塔等价性和莫里塔对偶性以及投影单元模子子类
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-04-05 DOI: 10.1007/s10485-024-09764-1
Ziba Fazelpour, Alireza Nasr-Isfahani
{"title":"Morita Equivalence and Morita Duality for Rings with Local Units and the Subcategory of Projective Unitary Modules","authors":"Ziba Fazelpour,&nbsp;Alireza Nasr-Isfahani","doi":"10.1007/s10485-024-09764-1","DOIUrl":"10.1007/s10485-024-09764-1","url":null,"abstract":"<div><p>We study Morita equivalence and Morita duality for rings with local units. We extend Auslander’s results on the theory of Morita equivalence and the Azumaya–Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya–Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568466","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
Grothendieck’s Vanishing and Non-vanishing Theorems in an Abstract Module Category 抽象模类中的格罗登第克消失和非消失定理
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-04-04 DOI: 10.1007/s10485-024-09767-y
Divya Ahuja, Surjeet Kour
{"title":"Grothendieck’s Vanishing and Non-vanishing Theorems in an Abstract Module Category","authors":"Divya Ahuja,&nbsp;Surjeet Kour","doi":"10.1007/s10485-024-09767-y","DOIUrl":"10.1007/s10485-024-09767-y","url":null,"abstract":"<div><p>In this article, we prove Grothendieck’s Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let <i>k</i> be a field of characteristic zero and <span>({mathscr {S}}_{k})</span> be a strongly locally noetherian <i>k</i>-linear Grothendieck category. For a commutative noetherian <i>k</i>-algebra <i>R</i>, let <span>({mathscr {S}}_R)</span> denote the category of <i>R</i>-objects in <span>({mathscr {S}}_k)</span> obtained through a non-commutative base change by <i>R</i> of the abelian category <span>({mathscr {S}}_{k})</span>. First, we establish Grothendieck’s Vanishing Theorem for any object <span>({mathscr {M}})</span> in <span>({mathscr {S}}_{R})</span>. Further, if <i>R</i> is local and <span>({mathscr {S}}_{k})</span> is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object <span>({mathscr {M}})</span> in <span>({mathscr {S}}_R)</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09767-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568363","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
The Categorical Basis of Dynamical Entropy 动态熵的分类基础
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-03-08 DOI: 10.1007/s10485-024-09763-2
Suddhasattwa Das
{"title":"The Categorical Basis of Dynamical Entropy","authors":"Suddhasattwa Das","doi":"10.1007/s10485-024-09763-2","DOIUrl":"10.1007/s10485-024-09763-2","url":null,"abstract":"<div><p>Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system—which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075428","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
Idempotent Completions of n-Exangulated Categories n 外切范畴的幂等补全
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-02-08 DOI: 10.1007/s10485-023-09758-5
Carlo Klapproth, Dixy Msapato, Amit Shah
{"title":"Idempotent Completions of n-Exangulated Categories","authors":"Carlo Klapproth,&nbsp;Dixy Msapato,&nbsp;Amit Shah","doi":"10.1007/s10485-023-09758-5","DOIUrl":"10.1007/s10485-023-09758-5","url":null,"abstract":"<div><p>Suppose <span>((mathcal {C},mathbb {E},mathfrak {s}))</span> is an <i>n</i>-exangulated category. We show that the idempotent completion and the weak idempotent completion of <span>(mathcal {C})</span> are again <i>n</i>-exangulated categories. Furthermore, we also show that the canonical inclusion functor of <span>(mathcal {C})</span> into its (resp. weak) idempotent completion is <i>n</i>-exangulated and 2-universal among <i>n</i>-exangulated functors from <span>((mathcal {C},mathbb {E},mathfrak {s}))</span> to (resp. weakly) idempotent complete <i>n</i>-exangulated categories. Furthermore, we prove that if <span>((mathcal {C},mathbb {E},mathfrak {s}))</span> is <i>n</i>-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and <span>((n+2))</span>-angulated cases. However, our constructions recover the known structures in the established cases up to <i>n</i>-exangulated isomorphism of <i>n</i>-exangulated categories.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09758-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758334","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
Operads, Operadic Categories and the Blob Complex 运算符、运算类和 Blob Complex
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-02-08 DOI: 10.1007/s10485-023-09759-4
Michael Batanin, Martin Markl
{"title":"Operads, Operadic Categories and the Blob Complex","authors":"Michael Batanin,&nbsp;Martin Markl","doi":"10.1007/s10485-023-09759-4","DOIUrl":"10.1007/s10485-023-09759-4","url":null,"abstract":"<div><p>We will show that the Morrison–Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary operadic categories and study some novel and interesting phenomena arising in this context.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09759-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758339","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
The Structure of Aisles and Co-aisles of t-Structures and Co-t-structures t 型结构和共 t 型结构的走道和共走道结构
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-02-06 DOI: 10.1007/s10485-023-09755-8
Aran Tattar
{"title":"The Structure of Aisles and Co-aisles of t-Structures and Co-t-structures","authors":"Aran Tattar","doi":"10.1007/s10485-023-09755-8","DOIUrl":"10.1007/s10485-023-09755-8","url":null,"abstract":"<div><p>Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category. Along the way, we also give an interpretation of these structures in the language of extriangulated categories.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09755-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758645","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
Algebraic Dynamical Systems in Machine Learning 机器学习中的代数动态系统
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-01-18 DOI: 10.1007/s10485-023-09762-9
Iolo Jones, Jerry Swan, Jeffrey Giansiracusa
{"title":"Algebraic Dynamical Systems in Machine Learning","authors":"Iolo Jones,&nbsp;Jerry Swan,&nbsp;Jeffrey Giansiracusa","doi":"10.1007/s10485-023-09762-9","DOIUrl":"10.1007/s10485-023-09762-9","url":null,"abstract":"<div><p>We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures for dynamic machine learning models (including recurrent neural networks, graph neural networks, and diffusion models) can be embedded. Considered in category theory, we also show that these algebraic models are a natural language for describing the compositionality of dynamic models. Furthermore, we propose that these models provide a template for the generalisation of the above dynamic models to learning problems on structured or non-numerical data, including ‘hybrid symbolic-numeric’ models.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09762-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139496018","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
The Stone Representations for Generalized Continuous Posets 广义连续 Posets 的石头表示法
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-01-16 DOI: 10.1007/s10485-023-09761-w
Ao Shen, Qingguo Li
{"title":"The Stone Representations for Generalized Continuous Posets","authors":"Ao Shen,&nbsp;Qingguo Li","doi":"10.1007/s10485-023-09761-w","DOIUrl":"10.1007/s10485-023-09761-w","url":null,"abstract":"<div><p>In this paper, we introduce the concepts of generalized continuous posets and present topological dualities for them. Moreover, we show that the category of generalized continuous posets and continuous morphisms is dually equivalent to the category of F-spaces and F-morphisms. In particular, some special cases are obtained, such as the topological representations for posets, domains, continuous lattices and join-semilattices.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139480389","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
Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality 非对称算子的兼容结构、马宁积和科斯祖尔对偶性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-01-10 DOI: 10.1007/s10485-023-09760-x
Huhu Zhang, Xing Gao, Li Guo
{"title":"Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality","authors":"Huhu Zhang,&nbsp;Xing Gao,&nbsp;Li Guo","doi":"10.1007/s10485-023-09760-x","DOIUrl":"10.1007/s10485-023-09760-x","url":null,"abstract":"<div><p>Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking a uniform approach, this paper presents an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly stronger restraints among the replicated copies. The linear compatibility is in Koszul duality to the total compatibility, while the matching compatibility is self dual. Further, each compatibility condition can be expressed in terms of either one or both of the two Manin square products. Finally it is shown that the operads defined by these compatibility conditions from the associative algebra and differential algebra are Koszul utilizing rewriting systems.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139409568","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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