Applied Categorical Structures最新文献

{"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}

{"title":"Diagrammatic Presentations of Enriched Monads and Varieties for a Subcategory of Arities","authors":"Rory B. B. Lucyshyn-Wright,&nbsp;Jason Parker","doi":"10.1007/s10485-023-09735-y","DOIUrl":"10.1007/s10485-023-09735-y","url":null,"abstract":"<div><p>The theory of <i>presentations</i> of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to <i>subcategories of arities</i> in recent work of Bourke–Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a <span>(mathscr {V})</span>-category <span>(mathscr {C})</span> in terms of <i>parametrized </i><span>(mathscr {J})</span>-<i>ary operations</i> and <i>diagrammatic equations</i> for a suitable subcategory of arities <span>(mathscr {J}hookrightarrow mathscr {C})</span>. On this basis we introduce the notions of <i>diagrammatic </i><span>(mathscr {J})</span>-<i>presentation</i> and <span>(mathscr {J})</span>-<i>ary variety</i>, and we show that the category of <span>(mathscr {J})</span>-ary varieties is dually equivalent to the category of <span>(mathscr {J})</span>-ary <span>(mathscr {V})</span>-monads. We establish several examples of diagrammatic <span>(mathscr {J})</span>-presentations and <span>(mathscr {J})</span>-ary varieties relevant in both mathematics and theoretical computer science, and we define the <i>sum</i> and <i>tensor product</i> of diagrammatic <span>(mathscr {J})</span>-presentations. We show that both <span>(mathscr {J})</span>-<i>relative monads</i> and <span>(mathscr {J})</span>-<i>pretheories</i> give rise to diagrammatic <span>(mathscr {J})</span>-presentations that directly describe their algebras. Using diagrammatic <span>(mathscr {J})</span>-presentations as a method of proof, we generalize the <i>pretheories-monads adjunction</i> of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff’s Galois connection between classes of algebras and sets of equations to the above setting.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50043178","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":"On the Structure of an Internal Groupoid","authors":"Nelson Martins-Ferreira","doi":"10.1007/s10485-023-09740-1","DOIUrl":"10.1007/s10485-023-09740-1","url":null,"abstract":"<div><p>The category of internal groupoids (in an arbitrary category) is shown to be equivalent to the full subcategory of so called involutive-2-links that are unital and associative.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09740-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50036814","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":"Admissibility of Localizations of Crossed Modules","authors":"Olivia Monjon,&nbsp;Jérôme Scherer,&nbsp;Florence Sterck","doi":"10.1007/s10485-023-09738-9","DOIUrl":"10.1007/s10485-023-09738-9","url":null,"abstract":"<div><p>The correspondence between the concept of conditional flatness and admissibility in the sense of Galois appears in the context of localization functors in any semi-abelian category admitting a fiberwise localization. It is then natural to wonder what happens in the category of crossed modules where fiberwise localization is not always available. In this article, we establish an equivalence between conditional flatness and admissibility in the sense of Galois (for the class of regular epimorphisms) for regular-epi localization functors. We use this equivalence to prove that nullification functors are admissible for the class of regular epimorphisms, even if the kernels of their localization morphisms are not acyclic.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09738-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50015273","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":"A Halmos–von Neumann Theorem for Actions of General Groups","authors":"Patrick Hermle,&nbsp;Henrik Kreidler","doi":"10.1007/s10485-023-09743-y","DOIUrl":"10.1007/s10485-023-09743-y","url":null,"abstract":"<div><p>We give a new categorical approach to the Halmos–von Neumann theorem for actions of general topological groups. As a first step, we establish that the categories of topological and measure-preserving irreducible systems with discrete spectrum are equivalent. This allows to prove the Halmos–von Neumann theorem in the framework of topological dynamics. We then use the Pontryagin and Tannaka–Krein duality theories to obtain classification results for topological and then measure-preserving systems with discrete spectrum. As a byproduct, we obtain a complete isomorphism invariant for compactifications of a fixed topological group.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09743-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41765715","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":"Correction: Presenting Quotient Locales","authors":"Graham Manuell","doi":"10.1007/s10485-023-09745-w","DOIUrl":"10.1007/s10485-023-09745-w","url":null,"abstract":"","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09745-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43379554","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":"Coactions on (C^*)-Algebras and Universal Properties","authors":"Erik Bédos,&nbsp;S. Kaliszewski,&nbsp;John Quigg,&nbsp;Jonathan Turk","doi":"10.1007/s10485-023-09741-0","DOIUrl":"10.1007/s10485-023-09741-0","url":null,"abstract":"<div><p>It is well-known that the maximalization of a coaction of a locally compact group on a C*-algebra enjoys a universal property. We show how this important property can be deduced from a categorical framework by exploiting certain properties of the maximalization functor for coactions. We also provide a dual proof for the universal property of normalization of coactions.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50013755","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":"Deriving Dualities in Pointfree Topology from Priestley Duality","authors":"G. Bezhanishvili,&nbsp;S. Melzer","doi":"10.1007/s10485-023-09739-8","DOIUrl":"10.1007/s10485-023-09739-8","url":null,"abstract":"<div><p>There are several prominent duality results in pointfree topology. Hofmann–Lawson duality establishes that the category of continuous frames is dually equivalent to the category of locally compact sober spaces. This restricts to a dual equivalence between the categories of stably continuous frames and stably locally compact spaces, which further restricts to Isbell duality between the categories of compact regular frames and compact Hausdorff spaces. We show how to derive these dualities from Priestley duality for distributive lattices, thus shedding new light on these classic results.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42602443","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":"Extension of Topological Groupoids and Hurewicz Morphisms","authors":"Saikat Chatterjee,&nbsp;Praphulla Koushik","doi":"10.1007/s10485-023-09744-x","DOIUrl":"10.1007/s10485-023-09744-x","url":null,"abstract":"<div><p>In this paper, we introduce the notion of a topological groupoid extension and relate it to the already existing notion of a gerbe over a topological stack. We further study the properties of a gerbe over a Hurewicz (resp. Serre) stack.\u0000\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09744-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44009957","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":"Hopf Monads: A Survey with New Examples and Applications","authors":"Aryan Ghobadi","doi":"10.1007/s10485-023-09732-1","DOIUrl":"10.1007/s10485-023-09732-1","url":null,"abstract":"<div><p>We survey the theory of Hopf monads on monoidal categories, and present new examples and applications. As applications, we utilise this machinery to present a new theory of cross products, as well as analogues of the Fundamental Theorem of Hopf algebras and Radford’s biproduct Theorem for Hopf algebroids. Additionally, we describe new examples of Hopf monads which arise from Galois and Ore extensions of bialgebras. We also classify Lawvere theories whose corresponding monads on the category of sets and functions become Hopf, as well as Hopf monads on the poset of natural numbers.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09732-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41933969","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}