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Applied Categorical Structures最新文献

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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, A. Razafindrakoto, J. Walters-Wayland
{"title":"Semilattice Base Hierarchy for Frames and Its Topological Ramifications","authors":"G. Bezhanishvili, F. Dashiell, A. Razafindrakoto, J. Walters-Wayland","doi":"10.1007/s10485-024-09766-z","DOIUrl":"https://doi.org/10.1007/s10485-024-09766-z","url":null,"abstract":"<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>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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, Chao Zhang","doi":"10.1007/s10485-024-09779-8","DOIUrl":"https://doi.org/10.1007/s10485-024-09779-8","url":null,"abstract":"<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>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"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, José Rubén Maldonado-Herrera","doi":"10.1007/s10485-024-09774-z","DOIUrl":"https://doi.org/10.1007/s10485-024-09774-z","url":null,"abstract":"<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>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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":"https://doi.org/10.1007/s10485-024-09772-1","url":null,"abstract":"<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>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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":"https://doi.org/10.1007/s10485-024-09768-x","url":null,"abstract":"<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>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"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, Laura Scull, Jordan Watts","doi":"10.1007/s10485-024-09770-3","DOIUrl":"https://doi.org/10.1007/s10485-024-09770-3","url":null,"abstract":"","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"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":"https://doi.org/10.1007/s10485-024-09769-w","url":null,"abstract":"<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>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Codescent and Bicolimits of Pseudo-Algebras 伪代数的编码和双极限
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-04-09 DOI: 10.1007/s10485-024-09765-0
Axel Osmond
{"title":"Codescent and Bicolimits of Pseudo-Algebras","authors":"Axel Osmond","doi":"10.1007/s10485-024-09765-0","DOIUrl":"https://doi.org/10.1007/s10485-024-09765-0","url":null,"abstract":"<p>We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of codescent objects. After prerequisites on pseudomonads and their pseudo-algebras, we give a 2-dimensional Linton theorem reducing bicocompleteness of 2-categories of pseudo-algebras to existence of bicoequalizers of codescent objects. Finally we prove this condition to be fulfilled in the case of a bifinitary pseudomonad, ensuring bicocompleteness.</p>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568364","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
Morita Equivalence and Morita Duality for Rings with Local Units and the Subcategory of Projective Unitary Modules 带局部单元的环的莫里塔等价性和莫里塔对偶性以及投影单元模子子类
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-04-05 DOI: 10.1007/s10485-024-09764-1
Ziba Fazelpour, Alireza Nasr-Isfahani
{"title":"Morita Equivalence and Morita Duality for Rings with Local Units and the Subcategory of Projective Unitary Modules","authors":"Ziba Fazelpour, Alireza Nasr-Isfahani","doi":"10.1007/s10485-024-09764-1","DOIUrl":"https://doi.org/10.1007/s10485-024-09764-1","url":null,"abstract":"<p>We study Morita equivalence and Morita duality for rings with local units. We extend Auslander’s results on the theory of Morita equivalence and the Azumaya–Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya–Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules.\u0000</p>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568466","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
Grothendieck’s Vanishing and Non-vanishing Theorems in an Abstract Module Category 抽象模类中的格罗登第克消失和非消失定理
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-04-04 DOI: 10.1007/s10485-024-09767-y
Divya Ahuja, Surjeet Kour
{"title":"Grothendieck’s Vanishing and Non-vanishing Theorems in an Abstract Module Category","authors":"Divya Ahuja, Surjeet Kour","doi":"10.1007/s10485-024-09767-y","DOIUrl":"https://doi.org/10.1007/s10485-024-09767-y","url":null,"abstract":"<p>In this article, we prove Grothendieck’s Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let <i>k</i> be a field of characteristic zero and <span>({mathscr {S}}_{k})</span> be a strongly locally noetherian <i>k</i>-linear Grothendieck category. For a commutative noetherian <i>k</i>-algebra <i>R</i>, let <span>({mathscr {S}}_R)</span> denote the category of <i>R</i>-objects in <span>({mathscr {S}}_k)</span> obtained through a non-commutative base change by <i>R</i> of the abelian category <span>({mathscr {S}}_{k})</span>. First, we establish Grothendieck’s Vanishing Theorem for any object <span>({mathscr {M}})</span> in <span>({mathscr {S}}_{R})</span>. Further, if <i>R</i> is local and <span>({mathscr {S}}_{k})</span> is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object <span>({mathscr {M}})</span> in <span>({mathscr {S}}_R)</span>.</p>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568363","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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