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Applied Categorical Structures最新文献

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Dagger Categories and the Complex Numbers: Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions 匕首范畴与复数：有限维希尔伯特空间范畴的公理与线性收缩
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-05-26 DOI: 10.1007/s10485-025-09803-5
Matthew Di Meglio, Chris Heunen
{"title":"Dagger Categories and the Complex Numbers: Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions","authors":"Matthew Di Meglio,&nbsp;Chris Heunen","doi":"10.1007/s10485-025-09803-5","DOIUrl":"10.1007/s10485-025-09803-5","url":null,"abstract":"<div><p>We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and linear contractions in terms of simple category-theoretic structures and properties that do not refer to norms, continuity, or real numbers. This builds on Heunen, Kornell, and Van der Schaaf’s easier characterisation of the category of all Hilbert spaces and linear contractions.\u0000\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09803-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144135469","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 Operadic Theory of Convexity 凸性的操作理论
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-05-24 DOI: 10.1007/s10485-025-09809-z
Redi Haderi, Cihan Okay, Walker H. Stern
{"title":"The Operadic Theory of Convexity","authors":"Redi Haderi,&nbsp;Cihan Okay,&nbsp;Walker H. Stern","doi":"10.1007/s10485-025-09809-z","DOIUrl":"10.1007/s10485-025-09809-z","url":null,"abstract":"<div><p>In this paper, we present an operadic characterization of convexity utilizing a PROP which governs convex structures and derive several convex Grothendieck constructions. Our main focus is a Grothendieck construction which simultaneously captures convex structures and monoidal structures on categories. Our proof of this Grothendieck construction makes heavy use of both our operadic characterization of convexity and operads governing monoidal structures. We apply these new tools to two key concepts: entropy in information theory and quantum contextuality in quantum foundations. In the former, we explain that Baez, Fritz, and Leinster’s categorical characterization of entropy has a more natural formulation in terms of continuous, convex-monoidal functors out of convex Grothendieck constructions; and in the latter we show that certain convex monoids used to characterize contextual distributions naturally arise as convex Grothendieck constructions.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144131476","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
Lawvere’s Frobenius Reciprocity, the Modular Connections of Grandis and Dilworth’s Abstract Principal Ideals Lawvere的Frobenius互易，Grandis的模连接和Dilworth的抽象主理想
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-05-16 DOI: 10.1007/s10485-025-09808-0
Amartya Goswami, Zurab Janelidze, Graham Manuell
{"title":"Lawvere’s Frobenius Reciprocity, the Modular Connections of Grandis and Dilworth’s Abstract Principal Ideals","authors":"Amartya Goswami,&nbsp;Zurab Janelidze,&nbsp;Graham Manuell","doi":"10.1007/s10485-025-09808-0","DOIUrl":"10.1007/s10485-025-09808-0","url":null,"abstract":"<div><p>The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09808-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144074010","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
Hardly Groupoids and Internal Categories in Gumm Categories Gumm分类中的hard群和内部分类
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-04-05 DOI: 10.1007/s10485-025-09807-1
Dominique Bourn
{"title":"Hardly Groupoids and Internal Categories in Gumm Categories","authors":"Dominique Bourn","doi":"10.1007/s10485-025-09807-1","DOIUrl":"10.1007/s10485-025-09807-1","url":null,"abstract":"<div><p>Under the name of hardly groupoid, we investigate the (internal) categories in which any morphism is both monomorphic and epimorphic. It is the case for any internal category in a congruence modular variety. In the more general context of Gumm categories, we explore the large diversity of situation determined by this notion.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143784215","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
A Categorical Characterization of Quantum Projective (mathbb {Z})-spaces 量子射影(mathbb {Z}) -空间的范畴表征
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-03-29 DOI: 10.1007/s10485-025-09806-2
Izuru Mori, Adam Nyman
{"title":"A Categorical Characterization of Quantum Projective (mathbb {Z})-spaces","authors":"Izuru Mori,&nbsp;Adam Nyman","doi":"10.1007/s10485-025-09806-2","DOIUrl":"10.1007/s10485-025-09806-2","url":null,"abstract":"<div><p>In this paper we study a generalization of the notion of AS-regularity for connected <span>({mathbb Z})</span>-algebras defined in Mori and Nyman (J Pure Appl Algebra, 225(9), 106676, 2021). Our main result is a characterization of those categories equivalent to noncommutative projective schemes associated to right coherent regular <span>({mathbb Z})</span>-algebras, which we call quantum projective <span>({mathbb Z})</span>-spaces in this paper. As an application, we show that smooth quadric hypersurfaces and the standard noncommutative smooth quadric surfaces studied in Smith and Van den Bergh (J Noncommut Geom 7(3), 817–856, 2013) , Mori and Ueyama (J Noncommut Geom, 15(2), 489–529, 2021) have right noetherian AS-regular <span>({mathbb Z})</span>-algebras as homogeneous coordinate algebras. In particular, the latter are thus noncommutative <span>({mathbb P}^1times {mathbb P}^1)</span> [in the sense of Van den Bergh (Int Math Res Not 17:3983–4026, 2011)].</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143735430","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
A Characterisation for the Category of Hilbert Spaces 希尔伯特空间类别的特征描述
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-03-22 DOI: 10.1007/s10485-025-09805-3
Stephen Lack, Shay Tobin
{"title":"A Characterisation for the Category of Hilbert Spaces","authors":"Stephen Lack,&nbsp;Shay Tobin","doi":"10.1007/s10485-025-09805-3","DOIUrl":"10.1007/s10485-025-09805-3","url":null,"abstract":"<div><p>The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Solèr’s theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quarternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quarternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09805-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143667977","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
Sh(B)-Valued Models of ((kappa ,kappa ))-Coherent Categories Sh(B)- ((kappa ,kappa )) -相干范畴的值模型
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-03-11 DOI: 10.1007/s10485-025-09804-4
Kristóf Kanalas
{"title":"Sh(B)-Valued Models of ((kappa ,kappa ))-Coherent Categories","authors":"Kristóf Kanalas","doi":"10.1007/s10485-025-09804-4","DOIUrl":"10.1007/s10485-025-09804-4","url":null,"abstract":"<div><p>A basic technique in model theory is to name the elements of a model by introducing new constant symbols. We describe the analogous construction in the language of syntactic categories/sites. As an application we identify <span>(textbf{Set})</span>-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as \"<i>Sh</i>(<i>B</i>)-valued models\"). For the coherent fragment <span>(L_{omega omega }^g subseteq L_{omega omega })</span> this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to <span>(L_{kappa kappa }^g)</span> when <span>(kappa )</span> is weakly compact. We present some further applications: first, a <i>Sh</i>(<i>B</i>)-valued completeness theorem for <span>(L_{kappa kappa }^g)</span> (<span>(kappa )</span> is weakly compact), second, that <span>(mathcal {C}rightarrow textbf{Set} )</span> regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09804-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594659","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 Relations Relative to Factorisation Systems 与因式分解系统有关的双重关系类别
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-03-06 DOI: 10.1007/s10485-025-09799-y
Keisuke Hoshino, Hayato Nasu
{"title":"Double Categories of Relations Relative to Factorisation Systems","authors":"Keisuke Hoshino,&nbsp;Hayato Nasu","doi":"10.1007/s10485-025-09799-y","DOIUrl":"10.1007/s10485-025-09799-y","url":null,"abstract":"<div><p>We relativise double categories of relations to stable orthogonal factorisation systems. Furthermore, we present the characterisation of the relative double categories of relations in two ways. The first utilises a generalised comprehension scheme, and the second focuses on a specific class of vertical arrows defined solely double-categorically. We organise diverse classes of double categories of relations and correlate them with significant classes of factorisation systems. Our framework embraces double categories of spans and double categories of relations on regular categories, which we meticulously compare to existing work on the characterisations of bicategories and double categories of spans and relations.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143564445","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
Canonical extensions via fitted sublocales 通过拟合的子区域进行规范扩展
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-02-28 DOI: 10.1007/s10485-025-09802-6
Tomáš Jakl, Anna Laura Suarez
{"title":"Canonical extensions via fitted sublocales","authors":"Tomáš Jakl,&nbsp;Anna Laura Suarez","doi":"10.1007/s10485-025-09802-6","DOIUrl":"10.1007/s10485-025-09802-6","url":null,"abstract":"<div><p>We study restrictions of the correspondence between the lattice <span>(textsf{SE}(L))</span> of strongly exact filters, of a frame <i>L</i>, and the coframe <span>(mathcal {S}_o(L))</span> of fitted sublocales. In particular, we consider the classes of exact filters <span>(textsf{E}(L))</span>, regular filters <span>(textsf{R}(L))</span>, and the intersections <span>(mathcal {J}(textsf{CP}(L)))</span> and <span>(mathcal {J}(textsf{SO}(L)))</span> of completely prime and Scott-open filters, respectively. We show that all these classes of filters are sublocales of <span>(textsf{SE}(L))</span> and as such correspond to subcolocales of <span>(mathcal {S}_o(L))</span> with a concise description. The theory of polarities of Birkhoff is central to our investigations. We automatically derive universal properties for the said classes of filters by giving their descriptions in terms of polarities. The obtained universal properties strongly resemble that of the canonical extensions of lattices. We also give new equivalent definitions of subfitness in terms of the lattice of filters.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09802-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143513450","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
Lallement Functor is a Weak Right Multiadjoint 序函子是一个弱右多伴随子
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2025-02-08 DOI: 10.1007/s10485-025-09800-8
J. Climent Vidal, E. Cosme Llópez
{"title":"Lallement Functor is a Weak Right Multiadjoint","authors":"J. Climent Vidal,&nbsp;E. Cosme Llópez","doi":"10.1007/s10485-025-09800-8","DOIUrl":"10.1007/s10485-025-09800-8","url":null,"abstract":"<div><p>For a plural signature <span>(Sigma )</span> and with regard to the category <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span>, of naturally preordered idempotent <span>(Sigma )</span>-algebras and surjective homomorphisms, we define a contravariant functor <span>(textrm{Lsys}_{Sigma })</span> from <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> to <span>(textsf {Cat})</span>, the category of categories, that assigns to <span>({textbf {I}})</span> in <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> the category <span>({textbf {I}})</span>-<span>(textsf {LAlg}(Sigma ))</span>, of <span>({textbf {I}})</span>-semi-inductive Lallement systems of <span>(Sigma )</span>-algebras, and a covariant functor <span>((textsf {Alg}(Sigma ),{downarrow _{textsf {s}}}, cdot ))</span> from <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> to <span>(textsf {Cat})</span>, that assigns to <span>({textbf {I}})</span> in <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> the category <span>((textsf {Alg}(Sigma ),{downarrow _{textsf {s}}}, {textbf {I}}))</span>, of the coverings of <span>({textbf {I}})</span>, i.e., the ordered pairs <span>(({textbf {A}},f))</span> in which <span>({textbf {A}})</span> is a <span>(Sigma )</span>-algebra and <img> a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories <span>(int ^{textsf {NPIAlg}(Sigma )_{textsf {s}}}textrm{Lsys}_{Sigma })</span> and <span>(int _{textsf {NPIAlg}(Sigma )_{textsf {s}}}(textsf {Alg}(Sigma ),{downarrow _{textsf {s}}}, cdot ))</span>; define a functor <span>(mathfrak {L}_{Sigma })</span> from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09800-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143370031","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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