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Applied Categorical Structures最新文献

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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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
The Representation Theory of Brauer Categories I: Triangular Categories Brauer范畴的表示理论Ⅰ：三角范畴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-08-13 DOI: 10.1007/s10485-022-09689-7
Steven V Sam, Andrew Snowden
{"title":"The Representation Theory of Brauer Categories I: Triangular Categories","authors":"Steven V Sam,&nbsp;Andrew Snowden","doi":"10.1007/s10485-022-09689-7","DOIUrl":"10.1007/s10485-022-09689-7","url":null,"abstract":"<div><p>This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45896551","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: 34
Quotients of Span Categories that are Allegories and the Representation of Regular Categories 寓言跨范畴的商与正则范畴的表示
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-08-01 DOI: 10.1007/s10485-022-09687-9
S. N. Hosseini, A. R. Shir Ali Nasab, W. Tholen, L. Yeganeh
{"title":"Quotients of Span Categories that are Allegories and the Representation of Regular Categories","authors":"S. N. Hosseini,&nbsp;A. R. Shir Ali Nasab,&nbsp;W. Tholen,&nbsp;L. Yeganeh","doi":"10.1007/s10485-022-09687-9","DOIUrl":"10.1007/s10485-022-09687-9","url":null,"abstract":"<div><p>We consider the ordinary category <span>(mathsf {Span}({mathcal {C}}))</span> of (isomorphism classes of) spans of morphisms in a category <span>(mathcal {C})</span> with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of <span>(mathsf {Span}({mathcal {C}}))</span> to be an allegory. In particular, when <span>({mathcal {C}})</span> carries a pullback-stable, but not necessarily proper, <span>(({mathcal {E}},{mathcal {M}}))</span>-factorization system, we establish a quotient category <span>(mathsf {Span}_{{mathcal {E}}}({mathcal {C}}))</span> that is isomorphic to the category <span>(mathsf {Rel}_{{mathcal {M}}}({mathcal {C}}))</span> of <span>({mathcal {M}})</span>-relations in <span>({mathcal {C}})</span>, and show that it is a (unitary and tabular) allegory precisely when <span>({mathcal {M}})</span> is a class of monomorphisms in <span>({mathcal {C}})</span>. Without the restriction to monomorphisms, one can still find a least pullback-stable and composition-closed class <span>({mathcal {E}}_{bullet })</span> containing <span>(mathcal E)</span> such that <span>(mathsf {Span}_{{mathcal {E}}_{bullet }}({mathcal {C}}))</span> is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the 2-category of all regular categories.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49102939","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
On the Cocartesian Image of Preorders and Equivalence Relations in Regular Categories 关于正则范畴中序的Cocartesian映象和等价关系
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-07-25 DOI: 10.1007/s10485-022-09686-w
Dominique Bourn
{"title":"On the Cocartesian Image of Preorders and Equivalence Relations in Regular Categories","authors":"Dominique Bourn","doi":"10.1007/s10485-022-09686-w","DOIUrl":"10.1007/s10485-022-09686-w","url":null,"abstract":"<div><p>In a regular category <span>(mathbb {E})</span>, the direct image along a regular epimorphism <i>f</i> of a preorder is not a preorder in general. In <i>Set</i>, its best preorder approximation is then its cocartesian image above <i>f</i>. In a regular category, the existence of such a cocartesian image above <i>f</i> of a preorder <i>S</i> is actually equivalent to the existence of the supremum <span>(R[f]vee S)</span> among the preorders. We investigate here some conditions ensuring the existence of these cocartesian images or equivalently of these suprema. They apply to two very dissimilar contexts: any topos <span>(mathbb {E})</span> with suprema of countable chains of subobjects or any <i>n</i>-permutable regular category.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-022-09686-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42661708","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 Characterization of n-Gorenstein Tilting Comodules n-Gorenstein倾斜模的表征
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-07-19 DOI: 10.1007/s10485-022-09688-8
Yexuan Li, Hailou Yao
{"title":"A Characterization of n-Gorenstein Tilting Comodules","authors":"Yexuan Li,&nbsp;Hailou Yao","doi":"10.1007/s10485-022-09688-8","DOIUrl":"10.1007/s10485-022-09688-8","url":null,"abstract":"<div><p>The aim of this paper is to introduce the concept of <i>n</i>-Gorenstein tilting comodules and study its main properties. This concept generalizes the notion of <i>n</i>-tilting comodules of finite injective dimensions to the case of finite Gorenstein injective dimensions. As an application of our results, we discuss the problem of existence of complements to partial <i>n</i>-Gorenstein tilting comodules.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45533996","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
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} Completions of Leavitt-Path-Algebra Pullbacks 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} Completions of Leavitt-Path-Algebra Pullbacks
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2022-07-15 DOI: 10.1007/s10485-022-09685-x
A. Chirvasitu
{"title":"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} Completions of Leavitt-Path-Algebra Pullbacks","authors":"A. Chirvasitu","doi":"10.1007/s10485-022-09685-x","DOIUrl":"https://doi.org/10.1007/s10485-022-09685-x","url":null,"abstract":"","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44208267","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: 2
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