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Applied Categorical Structures最新文献

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Discrete Vertices in Simplicial Objects Internal to a Monoidal Category 一元范畴内的简单对象中的离散顶点
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-06-11 DOI: 10.1007/s10485-025-09812-4
Arne Mertens
{"title":"Discrete Vertices in Simplicial Objects Internal to a Monoidal Category","authors":"Arne Mertens","doi":"10.1007/s10485-025-09812-4","DOIUrl":"10.1007/s10485-025-09812-4","url":null,"abstract":"<div><p>We follow the work of Aguiar (Internal categories and quantum groups. PhD Thesis, Cornell University, 1997) on internal categories and introduce simplicial objects internal to a monoidal category as certain colax monoidal functors. Then we compare three approaches to equipping them with a discrete set of vertices. We introduce based colax monoidal functors and show that under suitable conditions they are equivalent to the templicial objects defined by Lowen and Mertens (Algebr Geom Topol, 2024). We also compare templicial objects to the enriched Segal precategories appearing in Lurie ((Infinity,2)-categories and the Goodwillie calculus I. Preprint at https://arxiv.org/abs/0905.0462v2), Simpson (Homotopy theory of higher categories. New mathematical monographs, Cambridge University Press, Cambridge, vol 19, p 634, 2012, https://doi.org/10.1017/CBO9780511978111) and Bacard (Theory Appl Categ 35:1227–1267, 2020), and show that they are equivalent for cartesian monoidal categories, but not in general.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145165171","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
A remark on the total simplicial set functor 关于全简单集函子的注解
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-06-09 DOI: 10.1007/s10485-025-09815-1
Danny Stevenson
{"title":"A remark on the total simplicial set functor","authors":"Danny Stevenson","doi":"10.1007/s10485-025-09815-1","DOIUrl":"10.1007/s10485-025-09815-1","url":null,"abstract":"<div><p>We prove that for any bisimplicial set <i>X</i>, the natural comparison map between the diagonal <i>dX</i> and the total simplicial set <i>TX</i> is a categorical equivalence in the sense of Joyal and Lurie.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09815-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145163866","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
0-Ideal Monad and Its Applications to Approach Spaces 0-理想单子及其在逼近空间中的应用
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-05-31 DOI: 10.1007/s10485-025-09813-3
Jinming Fang
{"title":"0-Ideal Monad and Its Applications to Approach Spaces","authors":"Jinming Fang","doi":"10.1007/s10485-025-09813-3","DOIUrl":"10.1007/s10485-025-09813-3","url":null,"abstract":"<div><p>In this paper, a notion of 0-ideals on a set is proposed and further it is shown that 0-ideals give rise to a power-enriched monad <span>(mathbb {Z})</span>=<span>(({textbf{Z}},m,e))</span> on the category of sets, namely <i>a 0-ideal monad</i>. As a first application, a new characterization of approach spaces is given by verifying that the category <span>({mathbb {Z}})</span>-<b>Mon</b> of <span>({mathbb {Z}})</span>-monoids is isomorphic to the category <b>App</b> of approach spaces. The second application consists of two components: (i) Based on 0-ideals, a concept of approach 0-convergence spaces is introduced. (ii) By using the Kleisli extension of <span>({textbf{Z}})</span>, the existence of an isomorphism between the category <b>AConv</b> of approach 0-convergence spaces and the category <span>({(mathbb {Z},2)})</span>-<b>Cat</b> of relational <span>({mathbb {Z}})</span>-algebras is verified. Then from the fact that <span>({mathbb {Z}})</span>-<b>Mon</b> and <span>({(mathbb {Z},2)})</span>-<b>Cat</b> are isomorphic, another new description of approach spaces is obtained by an isomorphism between <b>AConv</b> and <b>App</b>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145171202","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
Vopěnka’s Principle, Maximum Deconstructibility, and Singly-Generated Torsion Classes voponnka的原理，最大解构性和单一生成的扭转类
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-05-31 DOI: 10.1007/s10485-025-09814-2
Sean Cox
{"title":"Vopěnka’s Principle, Maximum Deconstructibility, and Singly-Generated Torsion Classes","authors":"Sean Cox","doi":"10.1007/s10485-025-09814-2","DOIUrl":"10.1007/s10485-025-09814-2","url":null,"abstract":"<div><p>Deconstructibility is an often-used sufficient condition on a class <span>(mathcal {C})</span> of modules that allows one to carry out homological algebra <i>relative to </i> <span>(mathcal {C})</span>. The principle <i>Maximum Deconstructibility (MD)</i> asserts that a certain necessary condition for a class to be deconstructible is also sufficient. MD implies, for example, that the classes of Gorenstein Projective modules, Ding Projective modules, their relativized variants, and all torsion classes are deconstructible over any ring. MD was known to follow from Vopěnka’s Principle and imply the existence of an <span>(omega _1)</span>-strongly compact cardinal. We prove that MD is equivalent to Vopěnka’s Principle, and to the assertion that each torsion class of abelian groups is generated by a single group within the class (yielding the converse of a theorem of Göbel and Shelah).</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09814-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145171830","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
Dagger Categories and the Complex Numbers: Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions 匕首范畴与复数：有限维希尔伯特空间范畴的公理与线性收缩
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-05-26 DOI: 10.1007/s10485-025-09803-5
Matthew Di Meglio, Chris Heunen
{"title":"Dagger Categories and the Complex Numbers: Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions","authors":"Matthew Di Meglio,&nbsp;Chris Heunen","doi":"10.1007/s10485-025-09803-5","DOIUrl":"10.1007/s10485-025-09803-5","url":null,"abstract":"<div><p>We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and linear contractions in terms of simple category-theoretic structures and properties that do not refer to norms, continuity, or real numbers. This builds on Heunen, Kornell, and Van der Schaaf’s easier characterisation of the category of all Hilbert spaces and linear contractions.\u0000\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09803-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144135469","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 Operadic Theory of Convexity 凸性的操作理论
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-05-24 DOI: 10.1007/s10485-025-09809-z
Redi Haderi, Cihan Okay, Walker H. Stern
{"title":"The Operadic Theory of Convexity","authors":"Redi Haderi,&nbsp;Cihan Okay,&nbsp;Walker H. Stern","doi":"10.1007/s10485-025-09809-z","DOIUrl":"10.1007/s10485-025-09809-z","url":null,"abstract":"<div><p>In this paper, we present an operadic characterization of convexity utilizing a PROP which governs convex structures and derive several convex Grothendieck constructions. Our main focus is a Grothendieck construction which simultaneously captures convex structures and monoidal structures on categories. Our proof of this Grothendieck construction makes heavy use of both our operadic characterization of convexity and operads governing monoidal structures. We apply these new tools to two key concepts: entropy in information theory and quantum contextuality in quantum foundations. In the former, we explain that Baez, Fritz, and Leinster’s categorical characterization of entropy has a more natural formulation in terms of continuous, convex-monoidal functors out of convex Grothendieck constructions; and in the latter we show that certain convex monoids used to characterize contextual distributions naturally arise as convex Grothendieck constructions.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144131476","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
Lawvere’s Frobenius Reciprocity, the Modular Connections of Grandis and Dilworth’s Abstract Principal Ideals Lawvere的Frobenius互易，Grandis的模连接和Dilworth的抽象主理想
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-05-16 DOI: 10.1007/s10485-025-09808-0
Amartya Goswami, Zurab Janelidze, Graham Manuell
{"title":"Lawvere’s Frobenius Reciprocity, the Modular Connections of Grandis and Dilworth’s Abstract Principal Ideals","authors":"Amartya Goswami,&nbsp;Zurab Janelidze,&nbsp;Graham Manuell","doi":"10.1007/s10485-025-09808-0","DOIUrl":"10.1007/s10485-025-09808-0","url":null,"abstract":"<div><p>The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09808-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144074010","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
Hardly Groupoids and Internal Categories in Gumm Categories Gumm分类中的hard群和内部分类
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-04-05 DOI: 10.1007/s10485-025-09807-1
Dominique Bourn
{"title":"Hardly Groupoids and Internal Categories in Gumm Categories","authors":"Dominique Bourn","doi":"10.1007/s10485-025-09807-1","DOIUrl":"10.1007/s10485-025-09807-1","url":null,"abstract":"<div><p>Under the name of hardly groupoid, we investigate the (internal) categories in which any morphism is both monomorphic and epimorphic. It is the case for any internal category in a congruence modular variety. In the more general context of Gumm categories, we explore the large diversity of situation determined by this notion.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143784215","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
A Categorical Characterization of Quantum Projective (mathbb {Z})-spaces 量子射影(mathbb {Z}) -空间的范畴表征
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-03-29 DOI: 10.1007/s10485-025-09806-2
Izuru Mori, Adam Nyman
{"title":"A Categorical Characterization of Quantum Projective (mathbb {Z})-spaces","authors":"Izuru Mori,&nbsp;Adam Nyman","doi":"10.1007/s10485-025-09806-2","DOIUrl":"10.1007/s10485-025-09806-2","url":null,"abstract":"<div><p>In this paper we study a generalization of the notion of AS-regularity for connected <span>({mathbb Z})</span>-algebras defined in Mori and Nyman (J Pure Appl Algebra, 225(9), 106676, 2021). Our main result is a characterization of those categories equivalent to noncommutative projective schemes associated to right coherent regular <span>({mathbb Z})</span>-algebras, which we call quantum projective <span>({mathbb Z})</span>-spaces in this paper. As an application, we show that smooth quadric hypersurfaces and the standard noncommutative smooth quadric surfaces studied in Smith and Van den Bergh (J Noncommut Geom 7(3), 817–856, 2013) , Mori and Ueyama (J Noncommut Geom, 15(2), 489–529, 2021) have right noetherian AS-regular <span>({mathbb Z})</span>-algebras as homogeneous coordinate algebras. In particular, the latter are thus noncommutative <span>({mathbb P}^1times {mathbb P}^1)</span> [in the sense of Van den Bergh (Int Math Res Not 17:3983–4026, 2011)].</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143735430","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
A Characterisation for the Category of Hilbert Spaces 希尔伯特空间类别的特征描述
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-03-22 DOI: 10.1007/s10485-025-09805-3
Stephen Lack, Shay Tobin
{"title":"A Characterisation for the Category of Hilbert Spaces","authors":"Stephen Lack,&nbsp;Shay Tobin","doi":"10.1007/s10485-025-09805-3","DOIUrl":"10.1007/s10485-025-09805-3","url":null,"abstract":"<div><p>The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Solèr’s theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quarternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quarternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09805-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143667977","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
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