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Applied Categorical Structures最新文献

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Presenting the Sierpinski Gasket in Various Categories of Metric Spaces 在各类度量空间中呈现西尔平斯基垫圈
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-07-30 DOI: 10.1007/s10485-024-09773-0
Jayampathy Ratnayake, Annanthakrishna Manokaran, Romaine Jayewardene, Victoria Noquez, Lawrence S. Moss
{"title":"Presenting the Sierpinski Gasket in Various Categories of Metric Spaces","authors":"Jayampathy Ratnayake,&nbsp;Annanthakrishna Manokaran,&nbsp;Romaine Jayewardene,&nbsp;Victoria Noquez,&nbsp;Lawrence S. Moss","doi":"10.1007/s10485-024-09773-0","DOIUrl":"10.1007/s10485-024-09773-0","url":null,"abstract":"<div><p>This paper studies presentations of the Sierpinski gasket as a final coalgebra for a functor on three categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, the second uses continuous maps, and the third uses Lipschitz maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. It was previously known that the Sierpinski gasket is bilipschitz equivalent (though not isomorhpic) to the final coalgebra of this functor in the category with short maps, and that final coalgebra is obtained by taking the completion of the initial algebra. In this paper, we prove that the Sierpiniski gasket itself is the final coalgebra in the category with continuous maps, though it does not occur as the completion of the initial algebra. In the Lipschitz setting, we show that the final coalgebra for this functor does not exist.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09773-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868291","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
Locally Coherent Exact Categories 局部相干的精确类别
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-07-26 DOI: 10.1007/s10485-024-09780-1
Leonid Positselski
{"title":"Locally Coherent Exact Categories","authors":"Leonid Positselski","doi":"10.1007/s10485-024-09780-1","DOIUrl":"10.1007/s10485-024-09780-1","url":null,"abstract":"<div><p>A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Št’ovíček. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09780-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771359","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
Universal Finite Functorial Semi-norms 通用有限函数半规范
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-07-19 DOI: 10.1007/s10485-024-09777-w
Clara Löh, Johannes Witzig
{"title":"Universal Finite Functorial Semi-norms","authors":"Clara Löh,&nbsp;Johannes Witzig","doi":"10.1007/s10485-024-09777-w","DOIUrl":"10.1007/s10485-024-09777-w","url":null,"abstract":"<div><p>Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the <span>(ell ^1)</span>-semi-norm. However, the <span>(ell ^1)</span>-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09777-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741478","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
Semilattice Base Hierarchy for Frames and Its Topological Ramifications 框架的半网格基础层次结构及其拓扑影响
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-07-10 DOI: 10.1007/s10485-024-09766-z
G. Bezhanishvili, F. Dashiell Jr., A. Razafindrakoto, J. Walters-Wayland
{"title":"Semilattice Base Hierarchy for Frames and Its Topological Ramifications","authors":"G. Bezhanishvili,&nbsp;F. Dashiell Jr.,&nbsp;A. Razafindrakoto,&nbsp;J. Walters-Wayland","doi":"10.1007/s10485-024-09766-z","DOIUrl":"10.1007/s10485-024-09766-z","url":null,"abstract":"<div><p>We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice <i>A</i>, we analyze the interval in the coframe of sublocales of the frame of downsets of <i>A</i> formed by all frames with the S-base <i>A</i>. We study various degrees of completeness of <i>A</i>, which generalize the concepts of extremally disconnected and basically disconnected frames. We introduce the concepts of D-bases and L-bases, as well as their bounded counterparts, and show how our results specialize and sharpen in these cases. Classic examples that are covered by our approach include zero-dimensional, completely regular, and coherent frames, allowing us to provide a new perspective on these well-studied classes of frames, as well as their spatial counterparts.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09766-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141571219","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
On String Algebras and the Cohen–Macaulay Auslander Algebras 论弦代数与科恩-麦考莱奥斯兰德代数
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-07-09 DOI: 10.1007/s10485-024-09779-8
Yu-Zhe Liu, Chao Zhang
{"title":"On String Algebras and the Cohen–Macaulay Auslander Algebras","authors":"Yu-Zhe Liu,&nbsp;Chao Zhang","doi":"10.1007/s10485-024-09779-8","DOIUrl":"10.1007/s10485-024-09779-8","url":null,"abstract":"<div><p>The Cohen–Macaulay Auslander algebra of an algebra <i>A</i> is defined as the endomorphism algebra of the direct sum of all indecomposable Gorenstein projective <i>A</i>-modules. The Cohen–Macaulay Auslander algebra of any string algebra is explicitly constructed in this paper. Moreover, it is shown that a class of special string algebras, which are called to be string algebras satisfying the G-<i>condition</i>, are representation-finite if and only if their Cohen–Macaulay Auslander algebras are representation-finite. As applications, it is proved that the derived representation type of gentle algebras coincide with their Cohen–Macaulay Auslander algebras.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570973","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
Internalizations of Decorated Bicategories via (pi _2)-Indexings 通过$$pi _2$$ -索引实现装饰二元范畴的内部化
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-06-22 DOI: 10.1007/s10485-024-09774-z
Juan Orendain, José Rubén Maldonado-Herrera
{"title":"Internalizations of Decorated Bicategories via (pi _2)-Indexings","authors":"Juan Orendain,&nbsp;José Rubén Maldonado-Herrera","doi":"10.1007/s10485-024-09774-z","DOIUrl":"10.1007/s10485-024-09774-z","url":null,"abstract":"<div><p>We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We consider structures on decorated 2-categories allowing us to formally implement arguments of sliding certain squares along vertical subdivisions in double categories. We call these structures <span>(pi _2)</span>-indexings. We present a construction associating, to every <span>(pi _2)</span>-indexing on a decorated 2-category, a length 1 double internalization.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09774-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509071","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
Parametrised Presentability Over Orbital Categories 轨道类别上的参数化呈现性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-06-07 DOI: 10.1007/s10485-024-09772-1
Kaif Hilman
{"title":"Parametrised Presentability Over Orbital Categories","authors":"Kaif Hilman","doi":"10.1007/s10485-024-09772-1","DOIUrl":"10.1007/s10485-024-09772-1","url":null,"abstract":"<div><p>In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09772-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552818","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
Pointless Parts of Completely Regular Frames 完全规则框架的无意义部分
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-06-04 DOI: 10.1007/s10485-024-09768-x
Richard N. Ball
{"title":"Pointless Parts of Completely Regular Frames","authors":"Richard N. Ball","doi":"10.1007/s10485-024-09768-x","DOIUrl":"10.1007/s10485-024-09768-x","url":null,"abstract":"<div><p>(Completely regular) locales generalize (Tychonoff) spaces; indeed, the passage from a locale to its spatial sublocale is a well understood coreflection. But a locale also possesses an equally important pointless sublocale, and with morphisms suitably restricted, the passage from a locale to its pointless sublocale is also a coreflection. Our main theorem is that every locale can be uniquely represented as a subdirect product of its pointless and spatial parts, again with suitably restricted projections. We then exploit this representation by showing that any locale is determined by (what may be described as) the placement of its points in its pointless part.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257350","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
Bicategories of Action Groupoids 动作群组的二范畴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-05-24 DOI: 10.1007/s10485-024-09770-3
Carla Farsi, Laura Scull, Jordan Watts
{"title":"Bicategories of Action Groupoids","authors":"Carla Farsi,&nbsp;Laura Scull,&nbsp;Jordan Watts","doi":"10.1007/s10485-024-09770-3","DOIUrl":"10.1007/s10485-024-09770-3","url":null,"abstract":"<div><p>We prove that the 2-category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences à la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors à la Roberts, and localising at all weak equivalences. These constructions generalise the known case of representable orbifold groupoids. We also show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two particularly nice forms of equivariant weak equivalences.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141102819","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 DG-Enhancement of ({text {D}}_{qc}(X)) with Applications in Deformation Theory $${text {D}}_{qc}(X)$$ 的 DG 增强及其在变形理论中的应用
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-05-08 DOI: 10.1007/s10485-024-09769-w
Francesco Meazzini
{"title":"A DG-Enhancement of ({text {D}}_{qc}(X)) with Applications in Deformation Theory","authors":"Francesco Meazzini","doi":"10.1007/s10485-024-09769-w","DOIUrl":"10.1007/s10485-024-09769-w","url":null,"abstract":"<div><p>It is well-known that DG-enhancements of the unbounded derived category <span>({text {D}}_{qc}(X))</span> of quasi-coherent sheaves on a scheme <i>X</i> are all equivalent to each other. Here we present an explicit model which leads to applications in deformation theory. In particular, we shall describe three models for derived endomorphisms of a quasi-coherent sheaf <span>(mathcal {F})</span> on a finite-dimensional Noetherian separated scheme (even if <span>(mathcal {F})</span> does not admit a locally free resolution). Moreover, these complexes are endowed with DG-Lie algebra structures, which we prove to control infinitesimal deformations of <span>(mathcal {F})</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09769-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930535","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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