Applied Categorical Structures最新文献

{"title":"Classification of Track (Bi)Categories via Group-Valued 3-Cocycles","authors":"Antonio M. Cegarra","doi":"10.1007/s10485-025-09825-z","DOIUrl":"10.1007/s10485-025-09825-z","url":null,"abstract":"<div><p>Track bicategories, where each hom-category is a groupoid, appear in various mathematical and physical contexts. In this paper, we establish a cohomological classification of track bicategories and track categories using group-valued 3-cocycles on small categories, formulated as lax functors into the one-object 3-category of groups. In the abelian case, this classification aligns with Baues-Wirsching cohomology for small categories with coefficients in natural systems, recovering previously known classification results.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050858","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}

{"title":"Fuzzy Simplicial Sets and Their Application to Geometric Data Analysis","authors":"Lukas Silvester Barth,&nbsp;Hannaneh Fahimi,&nbsp;Parvaneh Joharinad,&nbsp;Jürgen Jost,&nbsp;Janis Keck,&nbsp;Thomas Jan Mikhail","doi":"10.1007/s10485-025-09827-x","DOIUrl":"10.1007/s10485-025-09827-x","url":null,"abstract":"<div><p>In this article, we expand upon the concepts introduced in Spivak (Metric realization of fuzzy simplicial sets, 2009. http://www.dspivak.net/metric_realization090922.pdf) about the relationship between the category <span>(textbf{UM})</span> of uber metric spaces and the category <span>(textbf{sFuz})</span> of fuzzy simplicial sets. We show that fuzzy simplicial sets can be regarded as natural combinatorial generalizations of metric relations. Furthermore, we take inspiration from UMAP (McInnes et al, in: Umap: Uniform manifold approximation and projection for dimension reduction, 2018) to apply the theory to manifold learning, dimension reduction and data visualization, while refining some of their constructions to put the corresponding theory on a more solid footing. A generalization of the adjunction between <span>(textbf{UM})</span> and <span>(textbf{sFuz})</span> allows us to view the adjunctions used in both publications as special cases. Moreover, we derive an explicit description of colimits in <span>(textbf{UM})</span> and the realization functor <span>(text {Re}:textbf{sFuz}rightarrow textbf{UM})</span>, and show that <span>(textbf{UM})</span> can be embedded into <span>(textbf{sFuz})</span>. Furthermore, we prove analogous results for the category of extended-pseudo metric spaces <span>(textbf{EPMet})</span>. We also provide rigorous definitions of functors that make it possible to recursively merge sets of fuzzy simplicial sets and provide a description of the adjunctions between the category of truncated fuzzy simplicial sets and <span>(textbf{sFuz})</span>, which we relate to persistent homology. Combining those constructions, we can show a surprising connection between the well-known dimension reduction methods UMAP and Isomap (Tenenbaum et al. in Science 290(5500):2319–2323, 2000) and derive an alternative algorithm, which we call IsUMap, that combines some of the strengths of both methods. Additionally, we developed a new embedding method that allows to preserve clusters detected in the original metric space that we construct from the data. The visualization of the optimization process gives the user information, both about the inner-cluster distributions in the original metric space and their inter-cluster relations. We compare our new method with UMAP, Isomap and t-SNE on a series of low- and high-dimensional datasets and provide explanations for observed differences and improvements.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09827-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145028150","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}

{"title":"On Internal Categories and Crossed Objects in the Category of Monoids","authors":"Ilia Pirashvili","doi":"10.1007/s10485-025-09822-2","DOIUrl":"10.1007/s10485-025-09822-2","url":null,"abstract":"<div><p>In a previous work on quadratic algebras (Pirashvili in Glas Math J 61: 151–167, 2018), I constructed an internal category in the category of monoids, recalled in Sect. 3.2.1. Based on this, we introduce the notion of a crossed semi-bimodule in this paper. This new construction generalises the notion of a crossed semi-module, introduced independently by R. Street and A. Patchkoria, see Joyal (Macquarie Math Reports 860081, 1986) and Patchkoria (Georg Math J 5: 575–581, 1986) respectively. We also show that there is a one to one correspondence between crossed semi-bimodules and strict monoidal category structures on transformation categories satisfying the cc-condition, see Sects. 4 and 5.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144832120","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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"title":"Double Categories of Relations Relative to Factorisation Systems","authors":"Keisuke Hoshino,&nbsp;Hayato Nasu","doi":"10.1007/s10485-025-09799-y","DOIUrl":"10.1007/s10485-025-09799-y","url":null,"abstract":"<div><p>We relativise double categories of relations to stable orthogonal factorisation systems. Furthermore, we present the characterisation of the relative double categories of relations in two ways. The first utilises a generalised comprehension scheme, and the second focuses on a specific class of vertical arrows defined solely double-categorically. We organise diverse classes of double categories of relations and correlate them with significant classes of factorisation systems. Our framework embraces double categories of spans and double categories of relations on regular categories, which we meticulously compare to existing work on the characterisations of bicategories and double categories of spans and relations.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143564445","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}

{"title":"Canonical extensions via fitted sublocales","authors":"Tomáš Jakl,&nbsp;Anna Laura Suarez","doi":"10.1007/s10485-025-09802-6","DOIUrl":"10.1007/s10485-025-09802-6","url":null,"abstract":"<div><p>We study restrictions of the correspondence between the lattice <span>(textsf{SE}(L))</span> of strongly exact filters, of a frame <i>L</i>, and the coframe <span>(mathcal {S}_o(L))</span> of fitted sublocales. In particular, we consider the classes of exact filters <span>(textsf{E}(L))</span>, regular filters <span>(textsf{R}(L))</span>, and the intersections <span>(mathcal {J}(textsf{CP}(L)))</span> and <span>(mathcal {J}(textsf{SO}(L)))</span> of completely prime and Scott-open filters, respectively. We show that all these classes of filters are sublocales of <span>(textsf{SE}(L))</span> and as such correspond to subcolocales of <span>(mathcal {S}_o(L))</span> with a concise description. The theory of polarities of Birkhoff is central to our investigations. We automatically derive universal properties for the said classes of filters by giving their descriptions in terms of polarities. The obtained universal properties strongly resemble that of the canonical extensions of lattices. We also give new equivalent definitions of subfitness in terms of the lattice of filters.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09802-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143513450","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}