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The Differential Bundles of the Geometric Tangent Category of an Operad 算子几何切线范畴的差分束
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-08-19 DOI: 10.1007/s10485-024-09771-2
Marcello Lanfranchi
{"title":"The Differential Bundles of the Geometric Tangent Category of an Operad","authors":"Marcello Lanfranchi","doi":"10.1007/s10485-024-09771-2","DOIUrl":"10.1007/s10485-024-09771-2","url":null,"abstract":"<div><p>Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, <span>(mathscr {P})</span>-affine schemes can be defined as objects of the opposite of the category of algebras over an operad <span>(mathscr {P})</span>. An example is the opposite of the category of associative algebras. The category of operadic schemes of an operad carries a canonical tangent structure. This paper aims to initiate the study of the geometry of operadic affine schemes via this tangent category. For example, we expect the tangent structure over the opposite of the category of associative algebras to describe algebraic non-commutative geometry. In order to initiate such a program, the first step is to classify differential bundles, which are the analogs of vector bundles for differential geometry. In this paper, we prove that the tangent category of affine schemes of the enveloping operad <span>(mathscr {P}^{(A)})</span> over a <span>(mathscr {P})</span>-affine scheme <i>A</i> is precisely the slice tangent category over <i>A</i> of <span>(mathscr {P})</span>-affine schemes. We are going to employ this result to show that differential bundles over a <span>(mathscr {P})</span>-affine scheme <i>A</i> are precisely <i>A</i>-modules in the operadic sense.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208870","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
Semiseparable Functors and Conditions up to Retracts 半可分函数和条件直至撤回
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-08-13 DOI: 10.1007/s10485-024-09782-z
Alessandro Ardizzoni, Lucrezia Bottegoni
{"title":"Semiseparable Functors and Conditions up to Retracts","authors":"Alessandro Ardizzoni,&nbsp;Lucrezia Bottegoni","doi":"10.1007/s10485-024-09782-z","DOIUrl":"10.1007/s10485-024-09782-z","url":null,"abstract":"<div><p>In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09782-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208871","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
Homotopy Quotients and Comodules of Supercommutative Hopf Algebras 超交换霍普夫布拉斯的同调对数和协元
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-08-12 DOI: 10.1007/s10485-024-09781-0
Thorsten Heidersdorf, Rainer Weissauer
{"title":"Homotopy Quotients and Comodules of Supercommutative Hopf Algebras","authors":"Thorsten Heidersdorf,&nbsp;Rainer Weissauer","doi":"10.1007/s10485-024-09781-0","DOIUrl":"10.1007/s10485-024-09781-0","url":null,"abstract":"<div><p>We study model structures on the category of comodules of a supercommutative Hopf algebra <i>A</i> over fields of characteristic 0. Given a graded Hopf algebra quotient <span>(A rightarrow B)</span> satisfying some finiteness conditions, the Frobenius tensor category <span>({mathcal {D}})</span> of graded <i>B</i>-comodules with its stable model structure induces a monoidal model structure on <span>({mathcal {C}})</span>. We consider the corresponding homotopy quotient <span>(gamma : {mathcal {C}} rightarrow Ho {mathcal {C}})</span> and the induced quotient <span>({mathcal {T}} rightarrow Ho {mathcal {T}})</span> for the tensor category <span>({mathcal {T}})</span> of finite dimensional <i>A</i>-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in <span>(Ho {mathcal {T}})</span>. We apply these results in the <i>Rep</i>(<i>GL</i>(<i>m</i>|<i>n</i>))-case and study its homotopy category <span>(Ho {mathcal {T}})</span> associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of <span>(Ho{mathcal {T}})</span> by the negligible morphisms is again the representation category of a supergroup scheme.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09781-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938174","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
Topological Quantum Field Theories and Homotopy Cobordisms 拓扑量子场论与同调共线性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-07-31 DOI: 10.1007/s10485-024-09776-x
Fiona Torzewska
{"title":"Topological Quantum Field Theories and Homotopy Cobordisms","authors":"Fiona Torzewska","doi":"10.1007/s10485-024-09776-x","DOIUrl":"10.1007/s10485-024-09776-x","url":null,"abstract":"<div><p>We construct a category <span>({textrm{HomCob}})</span> whose objects are <i>homotopically 1-finitely generated</i> topological spaces, and whose morphisms are <i>cofibrant cospans</i>. Given a manifold submanifold pair (<i>M</i>, <i>A</i>), we prove that there exists functors into <span>({textrm{HomCob}})</span> from the full subgroupoid of the mapping class groupoid <span>(textrm{MCG}_{M}^{A})</span>, and from the full subgroupoid of the motion groupoid <span>(textrm{Mot}_{M}^{A})</span>, whose objects are homotopically 1-finitely generated. We also construct a family of functors <span>({textsf{Z}}_G:{textrm{HomCob}}rightarrow {textbf{Vect}})</span>, one for each finite group <i>G</i>. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf–Witten. Given a space <i>X</i>, we prove that <span>({textsf{Z}}_G(X))</span> can be expressed as the <span>({mathbb {C}})</span>-vector space with basis natural transformation classes of maps from <span>(pi (X,X_0))</span> to <i>G</i> for some finite representative set of points <span>(X_0subset X)</span>, demonstrating that <span>({textsf{Z}}_G)</span> is explicitly calculable.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09776-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868290","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
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
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