Applied Categorical Structures最新文献

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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
On n-unital and n-Mal’tsev categories 关于 n-unital 和 n-Mal'tsev 范畴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-10-15 DOI: 10.1007/s10485-024-09789-6
Dominique Bourn, Michael Hoefnagel
{"title":"On n-unital and n-Mal’tsev categories","authors":"Dominique Bourn,&nbsp;Michael Hoefnagel","doi":"10.1007/s10485-024-09789-6","DOIUrl":"10.1007/s10485-024-09789-6","url":null,"abstract":"<div><p>Inspired by some properties of the (dual of the) category of 2-nilpotent groups, we introduce the notion of 2-unital and 2-Mal’tsev categories which, in some sense, generalises the notion of unital and Mal’tsev categories, and we characterise their varietal occurrences. This is actually the first step of an inductive process which we begin to unfold.\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":"https://link.springer.com/content/pdf/10.1007/s10485-024-09789-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142438710","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
Homotopical Models for Metric Spaces and Completeness 度量空间的同托邦模型与完备性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-10-04 DOI: 10.1007/s10485-024-09788-7
Isaiah Dailey, Clara Huggins, Semir Mujevic, Chloe Shupe
{"title":"Homotopical Models for Metric Spaces and Completeness","authors":"Isaiah Dailey,&nbsp;Clara Huggins,&nbsp;Semir Mujevic,&nbsp;Chloe Shupe","doi":"10.1007/s10485-024-09788-7","DOIUrl":"10.1007/s10485-024-09788-7","url":null,"abstract":"<div><p>Categories enriched in the opposite poset of non-negative reals can be viewed as generalizations of metric spaces, known as Lawvere metric spaces. In this article, we develop model structures on the categories <span>({mathbb {R}_+text {-}textbf{Cat}})</span> and <span>({mathbb {R}_+text {-}textbf{Cat}}^textrm{sym})</span> of Lawvere metric spaces and symmetric Lawvere metric spaces, each of which captures different features pertinent to the study of metric spaces. More precisely, in the three model structures we construct, the fibrant–cofibrant objects are the extended metric spaces (in the usual sense), the Cauchy complete Lawvere metric spaces, and the Cauchy complete extended metric spaces, respectively. Finally, we show that two of these model structures are unique in a similar way to the canonical model structure on <span>(textbf{Cat})</span>.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409923","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 Commutant and Center of a Generalized Green Functor 广义绿色函数的换元和中心
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-09-25 DOI: 10.1007/s10485-024-09785-w
Sael Cruz Cabello
{"title":"The Commutant and Center of a Generalized Green Functor","authors":"Sael Cruz Cabello","doi":"10.1007/s10485-024-09785-w","DOIUrl":"10.1007/s10485-024-09785-w","url":null,"abstract":"<div><p>After fixing a commutative ring with unit <i>R</i>, we present the definition of <i>adequate category</i> and consider the category of <i>R</i>-linear functors from an adequate category to the category of <i>R</i>-modules. We endow this category of functors with a monoidal structure and study monoids (generalized Green functors) over it. For one of these generalized Green functors, we define two new monoids, its commutant and its center, and study some of their properties and relations between them. This work generalizes the article [3].</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413772","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 Characterization of Differential Bundles in Tangent Categories 切线范畴中微分束的特征
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-09-16 DOI: 10.1007/s10485-024-09786-9
Michael Ching
{"title":"A Characterization of Differential Bundles in Tangent Categories","authors":"Michael Ching","doi":"10.1007/s10485-024-09786-9","DOIUrl":"10.1007/s10485-024-09786-9","url":null,"abstract":"<div><p>A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of smooth vector bundle to the abstract setting. Here we provide a new characterization of differential bundles and show that, up to isomorphism, a differential bundle is determined by its projection map and zero section. We show how these results can be used to quickly identify differential bundles in various tangent categories.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249639","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
Diagrammatics for Comodule Monads 组合单子图解法
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-08-29 DOI: 10.1007/s10485-024-09778-9
Sebastian Halbig, Tony Zorman
{"title":"Diagrammatics for Comodule Monads","authors":"Sebastian Halbig,&nbsp;Tony Zorman","doi":"10.1007/s10485-024-09778-9","DOIUrl":"10.1007/s10485-024-09778-9","url":null,"abstract":"<div><p>We extend Willerton’s [24] graphical calculus for bimonads to comodule monads, a monadic interpretation of module categories over a monoidal category. As an application, we prove a version of Tannaka–Krein duality for these structures.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09778-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226626","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
Functorial Polar Functions in Compact Normal Joinfit Frames 紧凑法向 Joinfit 框架中的扇形极坐标函数
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-08-23 DOI: 10.1007/s10485-024-09783-y
Ricardo E. Carrera
{"title":"Functorial Polar Functions in Compact Normal Joinfit Frames","authors":"Ricardo E. Carrera","doi":"10.1007/s10485-024-09783-y","DOIUrl":"10.1007/s10485-024-09783-y","url":null,"abstract":"<div><p><span>(mathfrak {KNJ})</span> is the category of compact normal joinfit frames and frame homomorphisms. <span>(mathcal {P}F)</span> is the complete boolean algebra of polars of the frame <i>F</i>. A function <span>(mathfrak {X})</span> that assigns to each <span>(F in mathfrak {KNJ})</span> a subalgebra <span>(mathfrak {X}(F))</span> of <span>(mathcal {P}F)</span> that contains the complemented elements of <i>F</i> is a polar function. A polar function <span>(mathfrak {X})</span> is invariant (resp., functorial) if whenever <span>(phi : F longrightarrow H in mathfrak {KNJ})</span> is <span>(mathcal {P})</span>-essential (resp., skeletal) and <span>(p in mathfrak {X}(F))</span>, then <span>(phi (p)^{perp perp } in mathfrak {X}(H))</span>. <span>(phi : F longrightarrow H in mathfrak {KNJ})</span> is <span>(mathfrak {X})</span>-splitting if <span>(phi )</span> is <span>(mathcal {P})</span>-essential and whenever <span>(p in mathfrak {X}(F))</span>, then <span>(phi (p)^{perp perp })</span> is complemented in <i>H</i>. <span>(F in mathfrak {KNJ})</span> is <span>(mathfrak {X})</span>-projectable means that every <span>(p in mathfrak {X}(F))</span> is complemented. For a polar function <span>(mathfrak {X})</span> and <span>(F in mathfrak {KNJ})</span>, we construct the least <span>(mathfrak {X})</span>-splitting frame of <i>F</i>. Moreover, we prove that if <span>(mathfrak {X})</span> is a functorial polar function, then the class of <span>(mathfrak {X})</span>-projectable frames is a <span>(mathcal {P})</span>-essential monoreflective subcategory of <span>(mathfrak {KNJS})</span>, the category of <span>(mathfrak {KNJ})</span>-objects and skeletal maps (the case <span>(mathfrak {X}= mathcal {P})</span> is the result from Martínez and Zenk, which states that the class of strongly projectable <span>(mathfrak {KNJ})</span>-objects is a reflective subcategory of <span>(mathfrak {KNJS})</span>).</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208869","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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