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Non-Abelian Extensions of Groupoids and Their Groupoid Rings 类群的非阿贝尔扩展及其类群环
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-12-16 DOI: 10.1007/s10485-024-09795-8
Natã Machado, Johan Öinert, Stefan Wagner
{"title":"Non-Abelian Extensions of Groupoids and Their Groupoid Rings","authors":"Natã Machado,&nbsp;Johan Öinert,&nbsp;Stefan Wagner","doi":"10.1007/s10485-024-09795-8","DOIUrl":"10.1007/s10485-024-09795-8","url":null,"abstract":"<div><p>We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids <span>({mathcal {N}}rightarrow {mathcal {E}}rightarrow {mathcal {G}})</span> gives rise to a groupoid crossed product of <span>({mathcal {G}})</span> by the groupoid ring of <span>({mathcal {N}})</span> which recovers the groupoid ring of <span>({mathcal {E}})</span> up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09795-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142826110","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 Tangent Category Perspective on Connections in Algebraic Geometry
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-12-11 DOI: 10.1007/s10485-024-09796-7
G. S. H. Cruttwell, Jean-Simon Pacaud Lemay, Elias Vandenberg
{"title":"A Tangent Category Perspective on Connections in Algebraic Geometry","authors":"G. S. H. Cruttwell,&nbsp;Jean-Simon Pacaud Lemay,&nbsp;Elias Vandenberg","doi":"10.1007/s10485-024-09796-7","DOIUrl":"10.1007/s10485-024-09796-7","url":null,"abstract":"<div><p>There is an abstract notion of connection in any tangent category. In this paper, we show that when applied to the tangent category of affine schemes, this recreates the classical notion of a connection on a module (and similarly, in the tangent category of schemes, this recreates the notion of connection on a quasi-coherent sheaf of modules). By contrast, we also show that in the tangent category of algebras, there are no non-trivial connections.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142798557","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
Bi-accessible and Bipresentable 2-Categories
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-12-09 DOI: 10.1007/s10485-024-09794-9
Ivan Di Liberti, Axel Osmond
{"title":"Bi-accessible and Bipresentable 2-Categories","authors":"Ivan Di Liberti,&nbsp;Axel Osmond","doi":"10.1007/s10485-024-09794-9","DOIUrl":"10.1007/s10485-024-09794-9","url":null,"abstract":"<div><p>We develop a 2-dimensional version of accessibility and presentability compatible with the formalism of flat pseudofunctors. First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and cofinality; in particular we show that <span>(sigma )</span>-<i>filteredness</i> and <i>bifilteredness</i> are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of <i>bicompact</i> objects and <i>bifiltered</i> bicolimits. We then characterize them as categories of <i>flat pseudofunctors</i>. We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small <i>bilex</i> 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of finitary 2-monads on <span>(textbf{Cat})</span> are finitely bipresentable, which in particular captures the case of <span>(textbf{Lex})</span>, the 2-category of small lex categories. Invoking the technology of <i>lex-colimits</i>, we prove further that several 2-categories arising in categorical logic (<b>Reg, Ex, Coh, Ext, Adh, Pretop</b>) are also finitely bipresentable.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09794-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142798442","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
An Equivalence Between Two Models of (infty )-Categories of Enriched Presheaves 富预设类的（infty ）两个模型之间的等价性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-11-26 DOI: 10.1007/s10485-024-09792-x
Hadrian Heine
{"title":"An Equivalence Between Two Models of (infty )-Categories of Enriched Presheaves","authors":"Hadrian Heine","doi":"10.1007/s10485-024-09792-x","DOIUrl":"10.1007/s10485-024-09792-x","url":null,"abstract":"<div><p>Let <span>({{mathcal {O}}}rightarrow {text {BM}})</span> be a <span>({text {BM}})</span>-operad that exhibits an <span>(infty )</span>-category <span>({{mathcal {D}}})</span> as weakly bitensored over non-symmetric <span>(infty )</span>-operads <span>({{mathcal {V}}}rightarrow text {Ass }, {{mathcal {W}}}rightarrow text {Ass })</span> and <span>({{mathcal {C}}})</span> a <span>({{mathcal {V}}})</span>-enriched <span>(infty )</span>-precategory. We construct an equivalence </p><div><div><span>$$begin{aligned} text {Fun}_{text {Hin}}^{{mathcal {V}}}({{mathcal {C}}},{{mathcal {D}}}) simeq text {Fun}^{{mathcal {V}}}({{mathcal {C}}},{{mathcal {D}}}) end{aligned}$$</span></div></div><p>of <span>(infty )</span>-categories weakly right tensored over <span>({{mathcal {W}}})</span> between Hinich’s construction of <span>({{mathcal {V}}})</span>-enriched functors of Hinich (Adv Math 367:107129, 2020) and our construction of <span>({{mathcal {V}}})</span>-enriched functors of Heine (Adv Math 417:108941, 2023).\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09792-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714185","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
Operad Structures on the Species Composition of Two Operads 两个 Operad 的物种组成上的 Operad 结构
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-11-22 DOI: 10.1007/s10485-024-09793-w
Imen Rjaiba
{"title":"Operad Structures on the Species Composition of Two Operads","authors":"Imen Rjaiba","doi":"10.1007/s10485-024-09793-w","DOIUrl":"10.1007/s10485-024-09793-w","url":null,"abstract":"<div><p>We give an explicit description of two operad structures on the species composition <span>(textbf{p}circ textbf{q})</span>, where <span>(textbf{q})</span> is any given positive operad, and where <span>(textbf{p})</span> is the <span>({text{ NAP } })</span> operad, or a shuffle version of the magmatic operad <span>({text{ Mag } })</span>. No distributive law between <span>(textbf{p})</span> and <span>(textbf{q})</span> is assumed.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142691875","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
Dualizations of Approximations, (aleph _1)-Projectivity, and Vopěnka’s Principles 逼近的二元化、（aleph _1）投影性和沃培卡原理
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-11-02 DOI: 10.1007/s10485-024-09791-y
Asmae Ben Yassine, Jan Trlifaj
{"title":"Dualizations of Approximations, (aleph _1)-Projectivity, and Vopěnka’s Principles","authors":"Asmae Ben Yassine,&nbsp;Jan Trlifaj","doi":"10.1007/s10485-024-09791-y","DOIUrl":"10.1007/s10485-024-09791-y","url":null,"abstract":"<div><p>The approximation classes of modules that arise as components of cotorsion pairs are tied up by Salce’s duality. Here we consider general approximation classes of modules and investigate possibilities of dualization in dependence on closure properties of these classes. While some proofs are easily dualized, other dualizations require large cardinal principles, and some fail in ZFC, with counterexamples provided by classes of <span>(aleph _1)</span>-projective modules over non-perfect rings. For example, we show that the statement “each covering class of modules closed under homomorphic images is of the form <span>({mathrm{Gen,}}(M))</span> for a module <i>M</i>” is equivalent to Vopěnka’s Principle.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09791-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142565800","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
Generalized Multicategories: Change-of-Base, Embedding, and Descent 广义多类别：基础变化、嵌入和后裔
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-10-30 DOI: 10.1007/s10485-024-09775-y
Rui Prezado, Fernando Lucatelli Nunes
{"title":"Generalized Multicategories: Change-of-Base, Embedding, and Descent","authors":"Rui Prezado,&nbsp;Fernando Lucatelli Nunes","doi":"10.1007/s10485-024-09775-y","DOIUrl":"10.1007/s10485-024-09775-y","url":null,"abstract":"<div><p>Via the adjunction <span>( - *mathbbm {1} dashv mathcal V(mathbbm {1},-) :textsf {Span}({mathcal {V}}) rightarrow {mathcal {V}} text {-} textsf {Mat} )</span> and a cartesian monad <i>T</i> on an extensive category <span>( {mathcal {V}} )</span> with finite limits, we construct an adjunction <span>( - *mathbbm {1} dashv {mathcal {V}}(mathbbm {1},-) :textsf {Cat}(T,{mathcal {V}}) rightarrow ({overline{T}}, mathcal V)text{- }textsf{Cat} )</span> between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad <i>T</i> satisfies a suitable property, which holds for several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor <span>( - *mathbbm {1} :textsf {Set} rightarrow {mathcal {V}} )</span> is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures. These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09775-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142555266","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
Partial Algebras and Implications of (Weak) Matrix Properties 部分代数与（弱）矩阵属性的含义
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-10-26 DOI: 10.1007/s10485-024-09790-z
Michael Hoefnagel, Pierre-Alain Jacqmin
{"title":"Partial Algebras and Implications of (Weak) Matrix Properties","authors":"Michael Hoefnagel,&nbsp;Pierre-Alain Jacqmin","doi":"10.1007/s10485-024-09790-z","DOIUrl":"10.1007/s10485-024-09790-z","url":null,"abstract":"<div><p>Matrix properties are a type of property of categories which includes the ones of being Mal’tsev, arithmetical, majority, unital, strongly unital, and subtractive. Recently, an algorithm has been developed to determine implications <span>(textrm{M}Rightarrow _{textrm{lex}_*}textrm{N})</span> between them. We show here that this algorithm reduces to constructing a partial term corresponding to <span>(textrm{N})</span> from a partial term corresponding to <span>(textrm{M})</span>. Moreover, we prove that this is further equivalent to the corresponding implication between the weak versions of these properties, i.e., the one where only strong monomorphisms are considered instead of all monomorphisms.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09790-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142519128","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 Note on the Smash Product and Regular Associativity 关于粉碎积和正则关联性的说明
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-10-19 DOI: 10.1007/s10485-024-09787-8
Marco Grandis
{"title":"A Note on the Smash Product and Regular Associativity","authors":"Marco Grandis","doi":"10.1007/s10485-024-09787-8","DOIUrl":"10.1007/s10485-024-09787-8","url":null,"abstract":"<div><p>We want to study the smash product of pointed topological spaces, in an organic way and full generality, without relying on some ‘convenient subcategory’. The <i>n</i>-ary smash product has a ‘colax’ form of associativity, which supplies a categorical framework for the properties of this operation and its connection with the function spaces. Various concrete computations of smash products are given, including a large class of cases where associativity fails. Lax and colax monoidal structures are unusual and interesting, in category theory. Some parts of this note will be obvious to a topologist and others to a categorist, in order to take into account both backgrounds.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142451017","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
Acyclicity Conditions on Pasting Diagrams 粘贴图表的循环条件
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-10-15 DOI: 10.1007/s10485-024-09784-x
Amar Hadzihasanovic, Diana Kessler
{"title":"Acyclicity Conditions on Pasting Diagrams","authors":"Amar Hadzihasanovic,&nbsp;Diana Kessler","doi":"10.1007/s10485-024-09784-x","DOIUrl":"10.1007/s10485-024-09784-x","url":null,"abstract":"<div><p>We study various acyclicity conditions on higher-categorical pasting diagrams in the combinatorial framework of regular directed complexes. We present an apparently weakest acyclicity condition under which the <span>(omega )</span>-category presented by a diagram shape is freely generated in the sense of polygraphs. We then consider stronger conditions under which this <span>(omega )</span>-category is equivalent to one obtained from an augmented directed chain complex in the sense of Steiner, or consists only of subsets of cells in the diagram. Finally, we study the stability of these conditions under the operations of pasting, suspensions, Gray products, joins and duals.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142438711","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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