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Applied Categorical Structures最新文献

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Unitless Frobenius Quantales Unitless Frobenius Quantales
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-12-27 DOI: 10.1007/s10485-022-09699-5
Cédric de Lacroix, Luigi Santocanale
{"title":"Unitless Frobenius Quantales","authors":"Cédric de Lacroix,&nbsp;Luigi Santocanale","doi":"10.1007/s10485-022-09699-5","DOIUrl":"10.1007/s10485-022-09699-5","url":null,"abstract":"<div><p>It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices <span>(M_n)</span> and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-022-09699-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45317354","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: 1
A Simplicial Category for Higher Correspondences 高等对应的一个简单范畴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-12-27 DOI: 10.1007/s10485-022-09705-w
Redi Haderi
{"title":"A Simplicial Category for Higher Correspondences","authors":"Redi Haderi","doi":"10.1007/s10485-022-09705-w","DOIUrl":"10.1007/s10485-022-09705-w","url":null,"abstract":"<div><p>In this work we propose a realization of Lurie’s prediction that inner fibrations <span>(p: X rightarrow A)</span> are classified by <i>A</i>-indexed diagrams in a “higher category” whose objects are <span>(infty )</span>-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and <span>(infty )</span>-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43134041","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: 1
A Pullback Diagram in the Coarse Category 粗类的回调图
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-12-27 DOI: 10.1007/s10485-022-09707-8
Elisa Hartmann
{"title":"A Pullback Diagram in the Coarse Category","authors":"Elisa Hartmann","doi":"10.1007/s10485-022-09707-8","DOIUrl":"10.1007/s10485-022-09707-8","url":null,"abstract":"<div><p>This paper studies the asymptotic product of two metric spaces. It is well defined if one of the spaces is visual or if both spaces are geodesic. In this case the asymptotic product is the pullback of a limit diagram in the coarse category. Using this product construction we can define a homotopy theory on coarse metric spaces in a natural way. We prove that all finite colimits exist in the coarse category.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47806464","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: 3
q-Tensor and Exterior Centers, Related Degrees and Capability q-张量与外心、相关度和能力
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-12-26 DOI: 10.1007/s10485-022-09701-0
Raimundo Bastos, Ricardo de Oliveira, Guram Donadze, Noraí Romeu Rocco
{"title":"q-Tensor and Exterior Centers, Related Degrees and Capability","authors":"Raimundo Bastos,&nbsp;Ricardo de Oliveira,&nbsp;Guram Donadze,&nbsp;Noraí Romeu Rocco","doi":"10.1007/s10485-022-09701-0","DOIUrl":"10.1007/s10485-022-09701-0","url":null,"abstract":"<div><p>We introduce intermediate commutators and study their degrees. We define <span>((q, {}))</span>-capable groups and prove that a group <i>G</i> is <span>((q, {}))</span>-capable if and only if <span>(Z^{wedge }_{(q, {})}(G)=1)</span>.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-022-09701-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47400970","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: 1
Coactions on C∗documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$C^*$$end{document}-Algebras and Universal Properties 关于C*documentclass[12pt]｛minimum｝usepackage｛amsmath｝usapackage｛wasysym｝use package｛amsfonts｝usepackage｛amssymb｝userpackage｛amsbsy｝ usepackage{mathrsfs｝ userpackage{upgeek｝setlength｛doddsidemargin｝｛-69pt｝ begin｛document｝$C^*$end{document}-Algebras和通用属性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-12-08 DOI: 10.1007/s10485-023-09741-0
Erik B'edos, S. Kaliszewski, John Quigg, Jonathan Turk
{"title":"Coactions on C∗documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$C^*$$end{document}-Algebras and Universal Properties","authors":"Erik B'edos, S. Kaliszewski, John Quigg, Jonathan Turk","doi":"10.1007/s10485-023-09741-0","DOIUrl":"https://doi.org/10.1007/s10485-023-09741-0","url":null,"abstract":"","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43457547","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
Semantic Factorization and Descent 语义分解与下降
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-11-15 DOI: 10.1007/s10485-022-09694-w
Fernando Lucatelli Nunes
{"title":"Semantic Factorization and Descent","authors":"Fernando Lucatelli Nunes","doi":"10.1007/s10485-022-09694-w","DOIUrl":"10.1007/s10485-022-09694-w","url":null,"abstract":"<div><p>Let <span>({mathbb {A}})</span> be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism <i>p</i> exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the <i>two-dimensional cokernel diagram</i> of <i>p</i> is up to isomorphism the same as the semantic factorization of <i>p</i>, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of <i>p</i> trivially hold whenever <i>p</i> has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on <i>p</i>, namely, to be an effective faithful morphism of the 2-category <span>({mathbb {A}})</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"30 6","pages":"1393 - 1433"},"PeriodicalIF":0.6,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-022-09694-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45437569","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: 4
2-Cartesian Fibrations I: A Model for (infty )-Bicategories Fibred in (infty )-Bicategories 2-笛卡儿纤颤I:一个模型 (infty )-分类纤维 (infty )-分类
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-09-28 DOI: 10.1007/s10485-022-09693-x
Fernando Abellán García, Walker H. Stern
{"title":"2-Cartesian Fibrations I: A Model for (infty )-Bicategories Fibred in (infty )-Bicategories","authors":"Fernando Abellán García,&nbsp;Walker H. Stern","doi":"10.1007/s10485-022-09693-x","DOIUrl":"10.1007/s10485-022-09693-x","url":null,"abstract":"<div><p>In this paper, we provide a notion of <span>(infty )</span>-bicategories fibred in <span>(infty )</span>-bicategories which we call <i>2-Cartesian fibrations</i>. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call <i>lean triangles</i>, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set <i>S</i>. Over the terminal scaled simplicial set, this provides a new model structure modeling <span>(infty )</span>-bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an <span>(infty )</span>-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of <span>({text {Set}}^+_{Delta })</span>-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"30 6","pages":"1341 - 1392"},"PeriodicalIF":0.6,"publicationDate":"2022-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50052459","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: 4
Matrix Taxonomy and Bourn Localization 矩阵分类与Bourn定位
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-09-21 DOI: 10.1007/s10485-022-09692-y
Michael Hoefnagel, Pierre-Alain Jacqmin
{"title":"Matrix Taxonomy and Bourn Localization","authors":"Michael Hoefnagel,&nbsp;Pierre-Alain Jacqmin","doi":"10.1007/s10485-022-09692-y","DOIUrl":"10.1007/s10485-022-09692-y","url":null,"abstract":"<div><p>In a recent paper (Hoefnagel et al. in Theory Appl Categ 38:737–790, 2022), an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices—the so called ‘matrix properties’. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal’tsev, majority and arithmetical categories. Using a computer implementation of our algorithm, we can display all such properties given by matrices of fixed dimensions, grouped according to their Bourn localizations, as well as the implications between them.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"30 6","pages":"1305 - 1340"},"PeriodicalIF":0.6,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46803515","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: 1
2-Limits and 2-Terminal Objects are too Different 2-极限和2-终端对象差别太大
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-09-08 DOI: 10.1007/s10485-022-09691-z
tslil clingman, Lyne Moser
{"title":"2-Limits and 2-Terminal Objects are too Different","authors":"tslil clingman,&nbsp;Lyne Moser","doi":"10.1007/s10485-022-09691-z","DOIUrl":"10.1007/s10485-022-09691-z","url":null,"abstract":"<div><p>In ordinary category theory, limits are known to be equivalent to terminal objects in the slice category of cones. In this paper, we prove that the 2-categorical analogues of this theorem relating 2-limits and 2-terminal objects in the various choices of slice 2-categories of 2-cones are false. Furthermore we show that, even when weakening the 2-cones to pseudo- or lax-natural transformations, or considering bi-type limits and bi-terminal objects, there is still no such correspondence.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"30 6","pages":"1283 - 1304"},"PeriodicalIF":0.6,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-022-09691-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45690923","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: 11
A Topological Duality for Monotone Expansions of Semilattices 半格单调展开的拓扑对偶性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-08-29 DOI: 10.1007/s10485-022-09690-0
Ismael Calomino, Paula Menchón, William J. Zuluaga Botero
{"title":"A Topological Duality for Monotone Expansions of Semilattices","authors":"Ismael Calomino,&nbsp;Paula Menchón,&nbsp;William J. Zuluaga Botero","doi":"10.1007/s10485-022-09690-0","DOIUrl":"10.1007/s10485-022-09690-0","url":null,"abstract":"<div><p>In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in S. Celani, L.J. González (Appl Categ Struct 28:853–875, 2020) for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies. \u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"30 6","pages":"1257 - 1282"},"PeriodicalIF":0.6,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-022-09690-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48148190","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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