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On the triangulated structure of stable monomorphism categories 稳定单态范畴的三角化结构
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-10-06 DOI: 10.1007/s10485-025-09842-y
Jonas Frank, Mathias Schulze
{"title":"On the triangulated structure of stable monomorphism categories","authors":"Jonas Frank,&nbsp;Mathias Schulze","doi":"10.1007/s10485-025-09842-y","DOIUrl":"10.1007/s10485-025-09842-y","url":null,"abstract":"<div><p>We investigate the triangulated structure of stable monomorphism categories (filtered chain categories) over a Frobenius category. The high degree of symmetry of linear quivers leads to a plethora of semiorthogonal decompositions into smaller categories of the same type. These form polygons of recollements, in which a full turn of mutations is a power of a particular auto-equivalence of the stable monomorphism category. A certain power of this auto-equivalence is the square of the suspension functor. We describe the infinite chains of adjoint pairs obtained from the polygons. As an application, we explicate the construction of Bondal and Kapranov for lifting representing objects of dualized hom-functors in our setup.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09842-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149098374","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
Convergence of Martingales via Enriched Dagger Categories 鞅的富匕首范畴收敛性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-10-04 DOI: 10.1007/s10485-026-09896-6
Paolo Perrone, Ruben Van Belle
{"title":"Convergence of Martingales via Enriched Dagger Categories","authors":"Paolo Perrone,&nbsp;Ruben Van Belle","doi":"10.1007/s10485-026-09896-6","DOIUrl":"10.1007/s10485-026-09896-6","url":null,"abstract":"<div><p>We provide a categorical proof of convergence for martingales and backward martingales in mean, using enriched category theory. The enrichment we use is in topological spaces, with their canonical closed monoidal structure, which encodes a version of pointwise convergence. We work in a topologically enriched dagger category of probability spaces and Markov kernels up to almost sure equality. In this category we can describe conditional expectations exactly as dagger-split idempotent morphisms, and filtrations can be encoded as directed nets of split idempotents, with their canonical partial order structure. As we show, every increasing (or decreasing) net of idempotents tends topologically to its supremum (or infimum). Random variables on a probability space form contravariant functors into categories of Hilbert and Banach spaces, which we can enrich topologically using the <span>(L^p)</span> norms. Martingales and backward martingales can be defined in terms of these functors. Since enriched functors preserve convergence of nets, we obtain convergence in the <span>(L^p)</span> norms. The convergence result for backward martingales indexed by an arbitrary net, in particular, seems to be new. By changing the functor, one can describe more general notions of conditional expectations and martingales, and if the functor is enriched, one automatically obtains a convergence result. For instance, one can recover the Bochner-based notion of vector-valued conditional expectation, and the convergence of martingales with values in an arbitrary Banach space. This work seems to be the first application of topologically enriched categories to analysis and probability in the literature. We hope that this enrichment, so often overlooked in the past, will be used in the future to obtain further convergence results categorically.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09896-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149098456","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
BV-Categories of Spacetime Interventions 时空干预的类别
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-09-23 DOI: 10.1007/s10485-026-09894-8
James Hefford, Matt Wilson
{"title":"BV-Categories of Spacetime Interventions","authors":"James Hefford,&nbsp;Matt Wilson","doi":"10.1007/s10485-026-09894-8","DOIUrl":"10.1007/s10485-026-09894-8","url":null,"abstract":"<div><p>We use the Chu construction to functorially build BV-categories from duoidal categories, demonstrating that candidate models of BV-logic can be cofreely constructed from a fragment of a model of Guglielmi’s sequencing operator. By using this construction to show that the strong Hyland envelope is a BV-category, we find a way to build a canonical model of spatio-temporal relationships between agents in spacetime from any symmetric monoidal category. The concrete physical interpretation of spacetime events in this model as intervention–context pairs resolves deficiencies in previous attempts to give a general categorical semantics to quantum supermaps.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149034533","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
Categorical Algebra of Conditional Probability 条件概率的范畴代数
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-08-29 DOI: 10.1007/s10485-026-09893-9
Mika Bohinen, Paolo Perrone
{"title":"Categorical Algebra of Conditional Probability","authors":"Mika Bohinen,&nbsp;Paolo Perrone","doi":"10.1007/s10485-026-09893-9","DOIUrl":"10.1007/s10485-026-09893-9","url":null,"abstract":"<div><p>In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and statistics. In this paper, we connect some ideas from categorical algebra, namely weakly cartesian functors and natural transformations, to the idea of conditioning in probability theory, using Markov categories and probability monads. First of all, we show that under some conditions, the monad associated to a Markov category with conditionals has a weakly cartesian functor and weakly cartesian multiplication. In particular, we show that this is the case for the Giry monad on standard Borel spaces. We then connect this theory to existing results on statistical experiments. We show that for deterministic statistical experiments, the so-called standard measure construction (which can be seen as a generalization of the “hypernormalizations” introduced by Jacobs) satisfies a universal property, allowing an equivalent definition which does not rely on the existence of conditionals.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148928100","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
Topological Representation, (mathrm {Delta }_{1})-Completion and Duality for Posets 拓扑表示，(mathrm {Delta }_{1}) -序集的补全和对偶性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-08-22 DOI: 10.1007/s10485-026-09895-7
Liping Zhang, Xiangnan Zhou, Qingguo Li
{"title":"Topological Representation, (mathrm {Delta }_{1})-Completion and Duality for Posets","authors":"Liping Zhang,&nbsp;Xiangnan Zhou,&nbsp;Qingguo Li","doi":"10.1007/s10485-026-09895-7","DOIUrl":"10.1007/s10485-026-09895-7","url":null,"abstract":"<div><p>In this paper, a topological representation for posets is presented by using irreducible Frink filters. Utilizing the topological representation, two particular <span>(Delta _{1})</span>-completions for posets are constructed. Furthermore, two categorical dualities are established for posets that differ essentially in their treatment of morphisms. The first, extending the Celani-González duality for meet-semilattices, interprets morphisms between topological spaces as certain relations, while keeping functions on the posets side. The second duality, in contrast, preserves the morphism structure on both sides as actual mappings, thereby yielding a fully functorial contravariant equivalence. Moreover, when restricted to the full subcategory of bounded distributive lattices, this second duality specializes to the classical Stone-type duality.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148837762","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
Enriched Categorical Semantics for Intuitionistic Subexponential Linear Logic (ISELL): Initial-Model Completeness 直觉次指数线性逻辑的丰富范畴语义：初始模型完备性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-08-20 DOI: 10.1007/s10485-026-09877-9
Carlos Ramírez Ovalle
{"title":"Enriched Categorical Semantics for Intuitionistic Subexponential Linear Logic (ISELL): Initial-Model Completeness","authors":"Carlos Ramírez Ovalle","doi":"10.1007/s10485-026-09877-9","DOIUrl":"10.1007/s10485-026-09877-9","url":null,"abstract":"<div><p>We give a <span>(mathcal {V})</span>-enriched categorical semantics for Intuitionistic Subexponential Linear Logic (ISELL), with <i>string-diagram</i> and numbered equational proofs in the main text for all one-step cut-reduction schemes. A model is a <span>(mathcal {V})</span>-enriched symmetric monoidal closed category (SMCC) equipped with a family of subexponential <i>coalgebra modalities</i> <span>((!^{a})_{ain (mathbb {E},preceq )})</span> and, whenever <span>(apreceq b)</span>, monoidal comonad morphisms <span>(theta ^{bRightarrow a}:!^{b}Rightarrow !^{a})</span> that preserve all available comonoid structure <span>((Delta ,e))</span>. This preservation yields an explicit “promotion prism” identity that semantically accounts for the subexponential side-condition. We build an initial enriched syntactic model by generators-and-relations and deduce completeness. We also prove a separation result showing that models without preserving comparisons cannot validate the same promotion behaviour.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09877-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148782948","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
Free Doubly-Infinitary Distributive Categories are Cartesian Closed 自由双无穷大分配范畴是笛卡尔闭范畴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-08-20 DOI: 10.1007/s10485-026-09887-7
Fernando Lucatelli Nunes, Matthijs Vákár
{"title":"Free Doubly-Infinitary Distributive Categories are Cartesian Closed","authors":"Fernando Lucatelli Nunes,&nbsp;Matthijs Vákár","doi":"10.1007/s10485-026-09887-7","DOIUrl":"10.1007/s10485-026-09887-7","url":null,"abstract":"<div><p>We study the composite free completion </p><div><div><span>$$ textsf{Dist}(mathcal {C}):=textsf{Fam}!bigl (textsf{Fam}(mathcal {C}^{op})^{op}bigr ), $$</span></div></div><p>obtained by first freely adjoining small products and then freely adjoining small coproducts. A natural pseudodistributive law equips this endo-pseudofunctor with a composite pseudomonad structure. Its pseudoalgebras are precisely the categories with small products and small coproducts in which small products distribute over small coproducts. We call such categories <i>doubly-infinitary distributive</i>. This condition is natural, but does not seem to have been systematically isolated in the literature. Thus <span>(textsf{Dist}(mathcal {C}))</span> is the free doubly-infinitary distributive category on <span>(mathcal {C})</span>. Our main result is that <span>(textsf{Dist}(mathcal {C}))</span> is <i>cartesian closed</i>. Finally, we compare doubly-infinitary distributivity with extensivity, ordinary infinitary distributivity, and cartesian closedness by means of separating examples.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09887-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148782949","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
Coalgebraic Models of (Omega )-Groups (Omega ) -群的共代数模型
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-08-12 DOI: 10.1007/s10485-026-09890-y
Maria Bevilacqua
{"title":"Coalgebraic Models of (Omega )-Groups","authors":"Maria Bevilacqua","doi":"10.1007/s10485-026-09890-y","DOIUrl":"10.1007/s10485-026-09890-y","url":null,"abstract":"<div><p>We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of coalgebraic models of algebraic theories endowed with an underlying structure of cocommutative Hopf algebra, and show that these categories are semi-abelian. We call them “categories of <span>(Omega )</span>-Hopf algebras”, since it is possible to characterize them as coalgebraic models of algebraic theories of <span>(Omega )</span>-groups in the sense of Higgins.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148751473","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
Unitary Magma Actions and Retraction Points 单一岩浆活动和收缩点
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-08-03 DOI: 10.1007/s10485-026-09889-5
Nelson Martins-Ferreira
{"title":"Unitary Magma Actions and Retraction Points","authors":"Nelson Martins-Ferreira","doi":"10.1007/s10485-026-09889-5","DOIUrl":"10.1007/s10485-026-09889-5","url":null,"abstract":"<div><p>We introduce a notion of action for unitary magmas and use it to classify a broad class of split extensions, called retraction points. Our approach is based on previous work on split extensions and semidirect products of unitary magmas, extending it to the more general setting in which the middle object of a split extension need not be in bijection with the Cartesian product of the kernel and cokernel objects. The construction is based on the notion of semibiproduct rather than the classical semidirect product of groups. Although this phenomenon does not occur in groups or, more generally, in associative semiabelian varieties, it appears in monoids through weakly Schreier extensions and in certain non-associative structures such as semi-left-loops. These results provide a broader framework for the study of split extensions of unitary magmas and related categorical structures.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148750768","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
Equivalence via Surjections 通过反射的等价
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-08-01 DOI: 10.1007/s10485-026-09875-x
Alexander Campbell, Tom Leinster
{"title":"Equivalence via Surjections","authors":"Alexander Campbell,&nbsp;Tom Leinster","doi":"10.1007/s10485-026-09875-x","DOIUrl":"10.1007/s10485-026-09875-x","url":null,"abstract":"<div><p>Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying <span>(textbf{A})</span> with <span>(textbf{B})</span> when there exists a strictly structure-preserving map <span>(textbf{A}rightarrow textbf{B})</span> that is genuinely (not just essentially) surjective in each dimension and faithful in the top dimension. We prove this principle for four types of structure: categories, monoidal categories, bicategories and double categories. The last of these theorems suggests that the right notion of equivalence between double categories is Campbell’s gregarious double equivalence, a conclusion also reached for different reasons in recent work of Moser, Sarazola and Verdugo.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09875-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148628567","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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