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A Recognition Criterion for Lax-Idempotent Pseudomonads 一类松弛幂等假单胞的识别准则
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-06-12 DOI: 10.1007/s10485-026-09871-1
John Bourke
{"title":"A Recognition Criterion for Lax-Idempotent Pseudomonads","authors":"John Bourke","doi":"10.1007/s10485-026-09871-1","DOIUrl":"10.1007/s10485-026-09871-1","url":null,"abstract":"<div><p>We describe a simple criterion which makes it easy to recognise when a pseudomonad is lax-idempotent. The criterion concerns the behaviour of colax bilimits of arrows—certain comma objects—and is easy to verify in examples. Building on this, we obtain a new characterisation of lax-idempotent pseudomonads on 2-categories with colax bilimits of arrows.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09871-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148237078","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
Abstract Cluster Structures 抽象簇结构
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-06-09 DOI: 10.1007/s10485-026-09876-w
Jan E. Grabowski, Sira Gratz
{"title":"Abstract Cluster Structures","authors":"Jan E. Grabowski,&nbsp;Sira Gratz","doi":"10.1007/s10485-026-09876-w","DOIUrl":"10.1007/s10485-026-09876-w","url":null,"abstract":"<div><p>We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an <i>abstract cluster structure</i>, which captures the essence of cluster mutation at a tropical level and show that cluster algebras, cluster varieties, cluster categories and surface models all have associated abstract cluster structures. For the first two classes, we also show that they can be constructed from abstract cluster structures. By defining a suitable notion of morphism of abstract cluster structures, we introduce a category of these and show that it has several desirable properties, such as initial and terminal objects and finite products and coproducts. We also prove that rooted cluster morphisms of cluster algebras give rise to morphisms of the associated abstract cluster structures, so that our framework includes a version of the extant category of cluster algebras. We can do more, however, because we can relate different types of representation of abstract cluster structures (cluster algebra, varieties, categories) directly via morphisms of their associated abstract cluster structures, even though no direct map from e.g. a cluster category to the associated cluster algebra is possible. In fact, we do much of the above in the setting of abstract quantum cluster structures, with some analysis of the difference between the category of these and that of the unquantized version. In order to show the relationship between abstract quantum cluster structures and quantum cluster algebras, we reformulate the usual construction of the latter in a way that is more amenable to our purposes and which we expect will be of independent interest and use.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09876-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148236606","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 Lattice of Smooth Sublocales as a Bruns–Lakser Completion 光滑亚域格的Bruns-Lakser补全
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-28 DOI: 10.1007/s10485-026-09873-z
Igor Arrieta, Anna Laura Suarez
{"title":"The Lattice of Smooth Sublocales as a Bruns–Lakser Completion","authors":"Igor Arrieta,&nbsp;Anna Laura Suarez","doi":"10.1007/s10485-026-09873-z","DOIUrl":"10.1007/s10485-026-09873-z","url":null,"abstract":"<div><p>We characterise the frame morphisms <span>(f:Lrightarrow M)</span> that lift to frame maps <span>(overline{f}:textsf{S}_b(L)rightarrow textsf{S}_b(M))</span>, where <span>(textsf{S}_b(L))</span> is the collection of joins of complemented sublocales of a frame <i>L</i>, or equivalently the Booleanization of the collection <span>(textsf{S}(L))</span> of all its sublocales. We do so by proving that <span>(textsf{S}_b(L))</span> is isomorphic to the Bruns–Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form <span>(mathfrak {c}(a)cap mathfrak {o}(b))</span> for <span>(a,bin L)</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09873-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148173590","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
From Coextensive Varieties to the Gaeta Topos 从共扩展品种到盖塔Topos
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-19 DOI: 10.1007/s10485-026-09872-0
William Zuluaga
{"title":"From Coextensive Varieties to the Gaeta Topos","authors":"William Zuluaga","doi":"10.1007/s10485-026-09872-0","DOIUrl":"10.1007/s10485-026-09872-0","url":null,"abstract":"<div><p>We analyze the relationship between coextensive varieties, central elements, and the Gaeta topos. Although the Gaeta topos captures a natural topos-theoretic notion of indecomposability, it does not in general classify all directly indecomposable algebras. We show that, in coextensive varieties, this discrepancy has a structural explanation: the geometric content of the Gaeta topos corresponds precisely to the absence of non-trivial central elements. Using the decomposition term available in every coextensive variety, we prove that central-freeness is expressible in the original algebraic language, yielding a geometric theory stable under interpretation in arbitrary toposes. Our main result establishes that the Gaeta topos classifies exactly this theory of central-free models. Equivalently, the Gaeta topology encodes the central decompositions of finitely presented algebras and forces the generic central decomposition induced by its covering families to become trivial.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148009146","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
Causally Disjoint Discs: Another (mathbb {E}_n)-Operad 因果分离盘：另一个(mathbb {E}_n) -操作
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-19 DOI: 10.1007/s10485-026-09874-y
Ryan Grady
{"title":"Causally Disjoint Discs: Another (mathbb {E}_n)-Operad","authors":"Ryan Grady","doi":"10.1007/s10485-026-09874-y","DOIUrl":"10.1007/s10485-026-09874-y","url":null,"abstract":"<div><p>Motivated by (perturbative) quantum observables in Lorentzian signature we define a new operad: the operad of causally disjoint disks. In order to describe this operad we use the orthogonal categories of Benini, Schenkel, and Woike and the prefactorization functor of Benini, Carmona, Grant-Stuart, and Schenkel. Along the way we extend these constructions to the topological setting, i.e., (multi-)categories enriched over spaces.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09874-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148009147","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
Invertibility and Parity in Symmetric Monoidal Categories 对称一元范畴的可逆性和宇称
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-14 DOI: 10.1007/s10485-026-09868-w
Nick Gurski, Niles Johnson
{"title":"Invertibility and Parity in Symmetric Monoidal Categories","authors":"Nick Gurski,&nbsp;Niles Johnson","doi":"10.1007/s10485-026-09868-w","DOIUrl":"10.1007/s10485-026-09868-w","url":null,"abstract":"<div><p>We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers <span>({pm })</span>1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09868-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147938445","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
Homological Lemmas for (Non-abelian) Group-Like Structures by Diagram Chasing in a Self-Dual Context 自对偶环境下（非阿贝尔）类群结构的图追的同调引理
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-09 DOI: 10.1007/s10485-026-09862-2
Kishan Kumar Dayaram, Amartya Goswami, Zurab Janelidze, Diana Rodelo, Tim Van der Linden
{"title":"Homological Lemmas for (Non-abelian) Group-Like Structures by Diagram Chasing in a Self-Dual Context","authors":"Kishan Kumar Dayaram,&nbsp;Amartya Goswami,&nbsp;Zurab Janelidze,&nbsp;Diana Rodelo,&nbsp;Tim Van der Linden","doi":"10.1007/s10485-026-09862-2","DOIUrl":"10.1007/s10485-026-09862-2","url":null,"abstract":"<div><p>Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper we show how the context of a ‘noetherian form’, recently introduced by the second and third authors, allows a self-dual treatment of these lemmas even in the case of non-abelian categories of group-like structures. This context covers a wide range of examples: module categories, the category of groups, of graded abelian groups, the categories of Lie algebras, of cocommutative Hopf algebras, the category of Heyting semilattices, of loops, the dual of the category of pointed sets, the category of modular/distributive lattices and modular connections, the category of sets and partial bijections, and many others. More generally, it includes all semi-abelian and Grandis exact categories.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09862-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147829970","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
Double Categories of Profunctors 函数的双范畴
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-09 DOI: 10.1007/s10485-026-09859-x
Yuto Kawase
{"title":"Double Categories of Profunctors","authors":"Yuto Kawase","doi":"10.1007/s10485-026-09859-x","DOIUrl":"10.1007/s10485-026-09859-x","url":null,"abstract":"<div><p>We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual double categories and, at the level of objects, up to isomorphism of enriched categories. Throughout the paper, we treat enrichment in a unital virtual double category rather than in a bicategory or a monoidal category, and, for consistency and better visualization of pasting diagrams, we adopt augmented virtual double categories as a fundamental language for double-categorical concepts.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2026-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09859-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147937926","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
Notions of Cauchy Completeness for Normed Categories 赋范范畴的柯西完备性概念
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-04 DOI: 10.1007/s10485-026-09865-z
Dirk Hofmann, Walter Tholen
{"title":"Notions of Cauchy Completeness for Normed Categories","authors":"Dirk Hofmann,&nbsp;Walter Tholen","doi":"10.1007/s10485-026-09865-z","DOIUrl":"10.1007/s10485-026-09865-z","url":null,"abstract":"<div><p>As already mentioned by Lawvere in his 1973 paper, the characterisation of Cauchy completeness of metric spaces in terms of representability of adjoint distributors amounts to the idempotent-split property of an ordinary category when the governing symmetric monoidal-closed category is changed from the extended real half-line to the category of sets. In this paper, for any commutative quantale <span>(mathcal {V})</span> taking the role of <span>([0,infty ])</span>, we extend these two characterisations of Lawvere-style completeness from ordinary to <span>(mathcal {V})</span>-normed categories, that is, to categories enriched in the category of <span>(mathcal {V})</span>-normed sets, i.e., of sets equipped with a <span>(mathcal {V})</span>-valued function. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important <span>(mathcal {V})</span>-normed categories.</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/s10485-026-09865-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147807590","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
Doubly Weak Double Categories 双弱双范畴
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2026-05-04 DOI: 10.1007/s10485-026-09863-1
Aaron David Fairbanks, Michael Shulman
{"title":"Doubly Weak Double Categories","authors":"Aaron David Fairbanks,&nbsp;Michael Shulman","doi":"10.1007/s10485-026-09863-1","DOIUrl":"10.1007/s10485-026-09863-1","url":null,"abstract":"<div><p>We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad—a structure with 2-cells of all possible double-categorical shapes—equipped with all possible composition operations, coherently. We also characterize them using “implicit” double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a “tidiness” condition to the double bicategories of Verity, or to the cubical bicategories of Garner.</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/s10485-026-09863-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147807591","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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