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On Generalized Morphisms Associated with Endofunctors of (C^*)-Algebras (C^*) -代数中与内函子相关的广义态射
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-30 DOI: 10.1007/s10485-026-09879-7
Georgii S. Makeev
{"title":"On Generalized Morphisms Associated with Endofunctors of (C^*)-Algebras","authors":"Georgii S. Makeev","doi":"10.1007/s10485-026-09879-7","DOIUrl":"10.1007/s10485-026-09879-7","url":null,"abstract":"<div><p>We introduce a class of good endofunctors of <span>(C^{*})</span>-algebras, endow it with a structure of a bimonoidal category, and define homotopies of natural transformations between such endofunctors. For every pair of <span>(C^{*})</span>-algebras and a good endofunctor, we construct a commutative monoid of generalized morphisms, and endow these monoids with a bilinear composition. This construction generalizes the homotopy category of asymptotic homomorphisms used in the definition of the Connes–Higson <i>E</i>-theory. We also introduce the notion of asymptotically adjoint good endofunctors, which has interesting applications to <i>E</i>-theory and <i>K</i>-homology.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615144","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
The Krull-Remak-Schmidt-Azumaya Theorem for Idempotent Complete Categories 幂等完备范畴的Krull-Remak-Schmidt-Azumaya定理
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-30 DOI: 10.1007/s10485-026-09888-6
Simion Breaz
{"title":"The Krull-Remak-Schmidt-Azumaya Theorem for Idempotent Complete Categories","authors":"Simion Breaz","doi":"10.1007/s10485-026-09888-6","DOIUrl":"10.1007/s10485-026-09888-6","url":null,"abstract":"<div><p>We prove a version of the Krull-Remak-Schmidt-Azumaya unique decomposition theorem in idempotent complete additive categories. In particular, this result applies to compactly generated triangulated categories. In addition, we provide an example of an object with the finite exchange property that does not have the exchange property.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09888-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615070","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
Cartesian Linearly Distributive Categories: Revisited 笛卡尔线性分配范畴：重访
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-26 DOI: 10.1007/s10485-026-09880-0
Rose Kudzman-Blais, Jean-Simon Pacaud Lemay
{"title":"Cartesian Linearly Distributive Categories: Revisited","authors":"Rose Kudzman-Blais,&nbsp;Jean-Simon Pacaud Lemay","doi":"10.1007/s10485-026-09880-0","DOIUrl":"10.1007/s10485-026-09880-0","url":null,"abstract":"<div><p>Linearly distributive categories (LDC) were introduced by Cockett and Seely to provide alternative categorical semantics for multiplicative linear logic. In contrast to Barr’s <span>(*)</span>-autonomous categories, LDCs take multiplicative conjunction and disjunction as primitive notions. Thus, a LDC is a category with two monoidal products that interact via linear distributors. A cartesian linearly distributive category (CLDC) is a LDC whose two monoidal products coincide with categorical products and coproducts. Initially, it was believed that CLDCs and distributive categories would coincide, but this was later found not to be the case. Consequently, the study of CLDCs was not pursued further at the time. With recent developments for and applications of LDCs, there has been renewed interest in CLDCs. This paper revisits CLDCs, demonstrating strong structural properties they all satisfy and investigating two key classes of examples: posetal distributive categories and semi-additive categories. Additionally, a previously assumed class of CLDCs, the Kleisli categories of exception monads of distributive categories, is re-examined and it is showed that they are not, in fact, CLDCs.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614188","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
Simplicial Effects and Weakly Associative Partial Groups 单纯效应与弱结合部分群
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-23 DOI: 10.1007/s10485-026-09878-8
Cihan Okay, Victor Castillo, Walker H. Stern
{"title":"Simplicial Effects and Weakly Associative Partial Groups","authors":"Cihan Okay,&nbsp;Victor Castillo,&nbsp;Walker H. Stern","doi":"10.1007/s10485-026-09878-8","DOIUrl":"10.1007/s10485-026-09878-8","url":null,"abstract":"<div><p>This paper introduces a broad generalization of effect algebras and of their multi-object counterparts effect algebroids, with the ultimate aim of providing a unified mathematical structure capable of capturing the simplicial, effect-algebra-like structures which arise from the theory of simplicial distributions. The key innovation of this paper is a weakened version of the associativity conditions satisfied by effect algebras and partial monoids, which allows us fit a much broader range of examples—in particular, many interesting classes of partial groups—into our framework. Indeed, within the rubric of weak associativity, simplicial effects and weakly associative partial groups arise as two extreme cases in the category of weak partial monoids. We conclude the paper by providing an example of a simplicial effect which is not an effect algebroid and showing that Gleason’s Theorem still holds for this simplicial effect.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09878-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614005","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
A Bitopological Duality for Some Subordination Boolean Algebras 一类隶属布尔代数的双拓扑对偶性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-23 DOI: 10.1007/s10485-026-09886-8
Sergio A. Celani
{"title":"A Bitopological Duality for Some Subordination Boolean Algebras","authors":"Sergio A. Celani","doi":"10.1007/s10485-026-09886-8","DOIUrl":"10.1007/s10485-026-09886-8","url":null,"abstract":"<div><p>S4-subordination algebras are a generalization of the closure algebras. In this paper, we give a topological representation for S4-subordination algebras by means of bitopological spaces <span>(langle X,tau ,tau _{S}rangle )</span>, where <span>(langle X,tau rangle )</span> is a Stone space and <span>(tau _{S})</span> is a topology that enables the characterization of the subordination relation. We apply this bitopological representation to give a characterization of S5-subordination algebras and lattice subordinations. We also show that there exists a bijective correspondence between congruence compatible with the subordination and certain closed subsets of the Stone space <span>(langle X,tau rangle )</span> that are also saturated sets of the space <span>(langle X,tau _{S}rangle )</span>. Additionally, we explore two types of morphisms between S4-subordination algebras: one based on Boolean homomorphisms and another based on meet-homomorphisms. Finally, we provide a topological representation for each type of morphism.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614003","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
Double Categories of Open Systems: The Cospan Approach 开放系统的双重范畴：Cospan方法
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-23 DOI: 10.1007/s10485-026-09885-9
John C. Baez
{"title":"Double Categories of Open Systems: The Cospan Approach","authors":"John C. Baez","doi":"10.1007/s10485-026-09885-9","DOIUrl":"10.1007/s10485-026-09885-9","url":null,"abstract":"<div><p>This is an overview of double categories of ‘open systems’: systems that can interact with their environment. We focus on the variable sharing paradigm, where we compose open systems by identifying variables. This paradigm is often implemented using structured or decorated cospans. We explain this approach using three main examples: open Petri nets, open dynamical systems, and open Petri nets with rates. We compare the virtues of structured and decorated cospan double categories, and study their common features. We show that any symmetric monoidal structured or decorated cospan double category comes with maps from two simpler double categories: its ‘exoskeleton’ and its ‘outer shell’. Finally, we study the concept of ‘hypergraph double category’, a kind of double category that should subsume structured and decorated cospans in a common framework for studying open systems in the variable sharing paradigm.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09885-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614004","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
Absolute Colimits 绝对Colimits
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-17 DOI: 10.1007/s10485-026-09847-1
Richard Garner, Ross Street
{"title":"Absolute Colimits","authors":"Richard Garner,&nbsp;Ross Street","doi":"10.1007/s10485-026-09847-1","DOIUrl":"10.1007/s10485-026-09847-1","url":null,"abstract":"<div><p>In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism <span>(alpha {:}M rightarrow {{mathscr {C}}}(F,Z))</span> to exhibit a functor <span>(Z{:}{{mathscr {A}}}rightarrow {{mathscr {C}}})</span> as an absolute <i>M</i>-weighted colimit of a functor <span>({F{:}{{mathscr {B}}}rightarrow {{mathscr {C}}}})</span>. We also review, with short proofs, the various criteria for the weight <i>M</i> itself to be absolute, in the sense that any <i>M</i>-weighted colimit is absolute. Finally, we prove that any absolute <i>M</i>-weighted colimit can be viewed as a colimit weighted by an absolute weight <span>(M')</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09847-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148466409","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
Rectangular Torsion Theories 矩形扭转理论
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-04 DOI: 10.1007/s10485-026-09867-x
Elena Caviglia, Zurab Janelidze, Luca Mesiti
{"title":"Rectangular Torsion Theories","authors":"Elena Caviglia,&nbsp;Zurab Janelidze,&nbsp;Luca Mesiti","doi":"10.1007/s10485-026-09867-x","DOIUrl":"10.1007/s10485-026-09867-x","url":null,"abstract":"<div><p>In this paper we introduce and study <i>rectangular torsion theories</i>, i.e. those torsion theories <span>((mathbb {C},mathcal {T},mathcal {F}))</span> with <span>(mathbb {C})</span> a pointed category, where the canonical functor <span>(mathbb {C}rightarrow mathcal {T}times mathcal {F})</span> is an equivalence of categories. We prove that rectangular torsion theories are pseudo-algebras for a suitable 2-monad on the 2-category of pointed categories. Interestingly, it turns out that this 2-monad is a 2-categorical counterpart of the monad for rectangular bands, i.e., non-empty semigroups satisfying <span>(xyz = xz)</span>. So rectangular torsion theories can be viewed as internal rectangular bands in the 2-category of pointed categories. We also show that a triple <span>((mathbb {C},mathcal {T},mathcal {F}))</span> is a rectangular torsion theory if and only if <span>(mathbb {C})</span> is a pointed category with full replete subcategories <span>(mathcal {T},mathcal {F})</span> such that: (i) every morphism between an object in <span>(mathcal {T})</span> and an object in <span>(mathcal {F})</span> (in either direction) is a zero morphism; (ii) any object in <span>(mathbb {C})</span> decomposes as a biproduct of an object in <span>(mathcal {T})</span> and an object in <span>(mathcal {F})</span>, and, all biproducts of an object in <span>(mathcal {T})</span> with an object in <span>(mathcal {F})</span> exist.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-026-09867-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148380665","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
Fundamental Category and Face Category of Stellar Complexes 恒星复合体的基类和面类
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-04 DOI: 10.1007/s10485-026-09882-y
Kohei Tanaka
{"title":"Fundamental Category and Face Category of Stellar Complexes","authors":"Kohei Tanaka","doi":"10.1007/s10485-026-09882-y","DOIUrl":"10.1007/s10485-026-09882-y","url":null,"abstract":"<div><p>We show that the fundamental category of a directed space decomposed into stellar strata by a poset (partially ordered set) is equivalent to its face category. As a consequence, we compute the fundamental category for two stellar stratifications on the classifying space of a small category.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148380666","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
Exponentiable Virtual Double Categories and Presheaves for Double Categories 可幂虚双范畴和双范畴的预轴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2026-07-01 DOI: 10.1007/s10485-026-09869-9
Nathanael Arkor
{"title":"Exponentiable Virtual Double Categories and Presheaves for Double Categories","authors":"Nathanael Arkor","doi":"10.1007/s10485-026-09869-9","DOIUrl":"10.1007/s10485-026-09869-9","url":null,"abstract":"<div><p>Given a pair of pseudo double categories <span>(mathbb {A})</span> and <span>(mathbb {B})</span>, the lax functors from <span>(mathbb {A})</span> to <span>(mathbb {B})</span>, along with their transformations, modules, and multimodulations, assemble into a virtual double category <span>(mathbb {L}{textbf {ax}} (mathbb {A}, mathbb {B}))</span>. We exhibit a universal property of this construction by observing that it arises naturally from the consideration of exponentiability for virtual double categories. In particular, we show that every pseudo double category is exponentiable as a virtual double category, whereby the virtual double category <span>(mathbb {L}{textbf {ax}} (mathbb {A}, mathbb {B}))</span> of lax functors arises as the virtual double category <span>(mathbb {M}{textbf {od}} (mathbb {B}^mathbb {A}))</span> of monads and modules in the exponential <span>(mathbb {B}^mathbb {A})</span>. We explore some consequences of this characterization, demonstrating that it leads to simple proofs of statements that heretofore required unwieldy computations. For instance, we deduce that the 2-category of pseudo double categories and lax functors is enriched in the 2-category of normal virtual double categories, and demonstrate that several aspects of the Yoneda theory of pseudo double categories—such as the correspondence between presheaves and discrete fibrations—are substantially simplified by this perspective.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"34 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148349653","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
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