{"title":"Metric Completions of Triangulated Categories from Finite Dimensional Algebras","authors":"Cyril Matoušek","doi":"10.1007/s10485-026-09857-z","DOIUrl":"10.1007/s10485-026-09857-z","url":null,"abstract":"<div><p>In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite representation type. In order to investigate completions of bounded derived categories of algebras of finite global dimension, we define image and preimage metrics under a triangulated functor and use them to induce a triangulated equivalence between two completions. Furthermore, for a given metric on a triangulated category we construct a new, closely related good metric called the improvement and compare the respective completions.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09857-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147807592","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":"Presheaves on Lax Double Functors; Or, Instances of Models of Double Theories","authors":"Kevin Carlson, Evan Patterson","doi":"10.1007/s10485-026-09866-y","DOIUrl":"10.1007/s10485-026-09866-y","url":null,"abstract":"<div><p>We introduce a notion of (co)presheaf on a lax double functor <i>X</i>, which we generally call an instance. In the terminology of double-categorical logic, a lax double functor valued in sets, possibly preserving finite products, is called a <i>model</i> of a double (Lawvere) theory. By varying the double theory, we uniformly define a well-behaved notion of instances of categories, profunctors, monads, monoidal categories, multicategories, and more, and we recover for instance the multifunctors into the category of sets in the last example. We show that instances of <i>X</i> can be described either in terms of modules from the terminal model <i>I</i> to <i>X</i>, satisfying an additional condition on triviality of the left action, or as loose natural transformations from <i>I</i> to <i>X</i>. We propose a notion of discrete opfibration between models of a double theory, establish a comprehensive factorization system, and prove an elements correspondence giving an equivalence between the category of instances of and the category of discrete opfibrations over a model <i>X</i>. We describe properties of the resulting categories of instances, relying on a “collage” construction which we characterize as a lax colimit of a model of a double theory. An appendix gives a detailed treatment of certain morphisms of lax functors relevant also for bicategory theory: (loose) transformations versus modules and modifications versus modulations.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147807593","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":"Length of Fully Faithful Framed Bicategories","authors":"Juan Orendain","doi":"10.1007/s10485-026-09861-3","DOIUrl":"10.1007/s10485-026-09861-3","url":null,"abstract":"<div><p>The length of a double category is a numerical invariant that quantifies the amount of ”work” required to rebuild the double category from its purely globular data. The minimal possible length is 1. It is conjectured that every framed bicategory realizes this minimal value. In this paper we confirm the conjecture for a subclass of framed bicategories: those whose unit squares are all cartesian/op-cartesian. We refer to such structures as fully-faithful/absolutely dense framed bicategories. The conjecture follows from a result relating the internal structure of fully faithful/absolutely dense framed bicategories to crossed products of decorated bicategories by <span>(pi _2)</span>-indexings.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147807589","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 Model Structure on the Category of (text{ A}_{infty })-Categories with Strict Morphisms","authors":"Mattia Ornaghi","doi":"10.1007/s10485-026-09870-2","DOIUrl":"10.1007/s10485-026-09870-2","url":null,"abstract":"<div><p>We prove that the category of (strictly unital) <span>(text{ A}_{infty })</span>-categories, linear over a commutative ring <i>R</i>, with strict <span>(text{ A}_{infty })</span>-morphisms has a cofibrantly generated model structure. In this model structure every object is fibrant and the cofibrant objects have cofibrant morphisms. As a consequence we prove that the semi-free <span>(text{ A}_{infty })</span>-categories (resp. resolutions) are cofibrant objects (resp. resolution) in this model structure.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09870-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147807594","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":"Simplicial Approach to Frobenius Algebras in the Category of Relations","authors":"Dominik Lachman","doi":"10.1007/s10485-026-09855-1","DOIUrl":"10.1007/s10485-026-09855-1","url":null,"abstract":"<div><p>Frobenius algebras in the category of sets and relations (<span>({textbf {Rel}})</span>) serve as a unifying framework for various algebraic and combinatorial structures, including groupoids, effect algebras, and abstract circles. Recently, a nerve construction of simplicial sets for Frobenius algebras in <span>({textbf {Rel}})</span> has been introduced. In this work, we investigate the lifting properties of these simplicial sets, linking them to the algebraic properties of Frobenius algebras. We introduce <span>(varepsilon )</span>-simplicial sets—simplicial sets with marked edges—that enable the representation of a broader class of structures, such as test spaces from quantum logic. Our main results focus on weakly saturated classes generated by cofibrations, corresponding to specific lifting problems. Furthermore, we provide a characterization of Frobenius algebras in <span>({textbf {Rel}})</span> within the framework of <span>(varepsilon )</span>-simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09855-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147737909","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}
J. Miguel Calderón, Alberto G. Raggi-Cárdenas, Itzel Rosas, Ramón H. Ruiz-Medina
{"title":"A Characterization of Groupoids in Terms of Their Category of (mathcal {C})-Sets","authors":"J. Miguel Calderón, Alberto G. Raggi-Cárdenas, Itzel Rosas, Ramón H. Ruiz-Medina","doi":"10.1007/s10485-026-09854-2","DOIUrl":"10.1007/s10485-026-09854-2","url":null,"abstract":"<div><p>A <span>(mathcal {C}-text {set})</span> is a functor from a category <span>(mathcal {C})</span> to the category of finite sets, and <span>(mathcal {C}-text {set})</span> denotes the category of such functors with natural transformations as morphisms. In this work, we prove that <span>(mathcal {C})</span> is a groupoid if and only if <span>(mathcal {C}-text {set})</span> has finitely many indecomposable objects, which is the categorical analog of transitive <i>G</i>-sets in classical group actions.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09854-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147606506","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":"Thomason Cohomology and Quillen’s Theorem A","authors":"Mehmet Kırtışoğlu, Ergün Yalçın","doi":"10.1007/s10485-026-09864-0","DOIUrl":"10.1007/s10485-026-09864-0","url":null,"abstract":"<div><p>Given a functor <span>(varphi : mathcal {C}rightarrow mathcal {D})</span> between small categories, Quillen showed that there is a homotopy equivalence <span>(kappa :mathop {textrm{hocolim}}limits _{mathcal {D}} N(varphi /-) rightarrow Nmathcal {C})</span>, where <span>(N(varphi /-))</span> is the functor that sends each object <i>d</i> of <span>(mathcal {D})</span> to the nerve of the comma category <span>(varphi /d)</span>. We show that the homotopy equivalence <span>(kappa )</span> induces an isomorphism on the Gabriel-Zisman cohomology of the associated simplicial sets. As a consequence, we obtain a version of Quillen’s Theorem A for the Thomason cohomology of categories. We also construct a spectral sequence converging to the Thomason cohomology of the Grothendieck construction <span>(int _{mathcal {D}} F)</span> of a functor <span>(F:mathcal {D}rightarrow textbf{Cat})</span> with arbitrary coefficients.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147606405","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}
Benjamin Merlin Bumpus, Wilmer Leal, James Fairbanks, Martti Karvonen, Frédéric Simard
{"title":"Towards a Unified Theory of Time-Varying Data","authors":"Benjamin Merlin Bumpus, Wilmer Leal, James Fairbanks, Martti Karvonen, Frédéric Simard","doi":"10.1007/s10485-026-09860-4","DOIUrl":"10.1007/s10485-026-09860-4","url":null,"abstract":"<div><p>What is a time-varying graph, a time-varying topological space, or, more generally, a mathematical structure that evolves over time? In this work, we lay the foundations for a general theory of temporal data by introducing <i>categories of narratives</i>. These are sheaves on posets of time intervals that encode snapshots of a temporal object along with the relationships between them. This theory satisfies five desiderata distilled from the burgeoning field of time-varying graphs: (D1) it defines both time-varying objects and their morphisms; (D2) it distinguishes between cumulative and persistent interpretations and provides principled methods for transitioning between them; (D3) it systematically lifts static notions to their temporal analogues; (D4) it is object agnostic; (D5) it integrates with theories of dynamical systems. To achieve this, we build upon existing categorical and sheaf-theoretic approaches to temporal graph theory, generalizing them to any category with limits and colimits. We also formalize tacit intuitions that, while present, often remain implicit in temporal graph theory. Beyond synthesizing and reformulating existing ideas in categorical language, we introduce sheaf-theoretic constructions and prove results that, to our knowledge, have not appeared in the temporal data literature—such as the adjunction between persistent and cumulative narratives. More importantly, we integrate these existing and novel elements into a consistent and coherent framework, setting the stage for a unified theory of time-varying data.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09860-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147606404","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":"Morphisms and Comorphisms of Sites I Double-Categories of Sites","authors":"Olivia Caramello, Axel Osmond","doi":"10.1007/s10485-026-09853-3","DOIUrl":"10.1007/s10485-026-09853-3","url":null,"abstract":"<div><p>We arrange morphisms and comorphisms of sites as the horizontal and vertical cells of a double category of sites; using the formalism of extensions and restrictions of presheaves, we explain how one can define a sheafification double functor from this double category to the quintet double category of Grothendieck topoi. We describe properties of this double functor and recover some classical results of topos theory through a new notion of <i>locally exact square</i>, generalizing exact squares in the presence of topologies. We also describe a 2-comonad on <span>(textbf{Cat})</span> for which lax morphisms of coalgebras are morphisms of sites and colax morphisms are comorphisms of sites, explaining the arrangement as a double category.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147606507","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":"Thomason’s Colimit Theorem for the Double Category of Elements","authors":"Andrew Gill, Maru Sarazola","doi":"10.1007/s10485-026-09858-y","DOIUrl":"10.1007/s10485-026-09858-y","url":null,"abstract":"<div><p>We show that, for any 2-category <span>(mathcal {C})</span> and 2-functor <span>(F:mathcal {C}^textrm{op}rightarrow underline{textrm{Cat}})</span>, the double category of elements <span>(iint _mathcal {C}F)</span> introduced by Grandis and Paré satisfies a version of Thomason’s colimit theorem; that is, there is a weak homotopy equivalence <span>(B{{,mathrm{textrm{hocolim}},}}Fsimeq B(iint _mathcal {C}F))</span>.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09858-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147560556","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}