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Applied Categorical Structures最新文献

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Operads, Operadic Categories and the Blob Complex 运算符、运算类和 Blob Complex
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-02-08 DOI: 10.1007/s10485-023-09759-4
Michael Batanin, Martin Markl
{"title":"Operads, Operadic Categories and the Blob Complex","authors":"Michael Batanin,&nbsp;Martin Markl","doi":"10.1007/s10485-023-09759-4","DOIUrl":"10.1007/s10485-023-09759-4","url":null,"abstract":"<div><p>We will show that the Morrison–Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary operadic categories and study some novel and interesting phenomena arising in this context.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09759-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758339","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 Structure of Aisles and Co-aisles of t-Structures and Co-t-structures t 型结构和共 t 型结构的走道和共走道结构
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-02-06 DOI: 10.1007/s10485-023-09755-8
Aran Tattar
{"title":"The Structure of Aisles and Co-aisles of t-Structures and Co-t-structures","authors":"Aran Tattar","doi":"10.1007/s10485-023-09755-8","DOIUrl":"10.1007/s10485-023-09755-8","url":null,"abstract":"<div><p>Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category. Along the way, we also give an interpretation of these structures in the language of extriangulated categories.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09755-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758645","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
Algebraic Dynamical Systems in Machine Learning 机器学习中的代数动态系统
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-01-18 DOI: 10.1007/s10485-023-09762-9
Iolo Jones, Jerry Swan, Jeffrey Giansiracusa
{"title":"Algebraic Dynamical Systems in Machine Learning","authors":"Iolo Jones,&nbsp;Jerry Swan,&nbsp;Jeffrey Giansiracusa","doi":"10.1007/s10485-023-09762-9","DOIUrl":"10.1007/s10485-023-09762-9","url":null,"abstract":"<div><p>We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures for dynamic machine learning models (including recurrent neural networks, graph neural networks, and diffusion models) can be embedded. Considered in category theory, we also show that these algebraic models are a natural language for describing the compositionality of dynamic models. Furthermore, we propose that these models provide a template for the generalisation of the above dynamic models to learning problems on structured or non-numerical data, including ‘hybrid symbolic-numeric’ models.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09762-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139496018","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 Stone Representations for Generalized Continuous Posets 广义连续 Posets 的石头表示法
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-01-16 DOI: 10.1007/s10485-023-09761-w
Ao Shen, Qingguo Li
{"title":"The Stone Representations for Generalized Continuous Posets","authors":"Ao Shen,&nbsp;Qingguo Li","doi":"10.1007/s10485-023-09761-w","DOIUrl":"10.1007/s10485-023-09761-w","url":null,"abstract":"<div><p>In this paper, we introduce the concepts of generalized continuous posets and present topological dualities for them. Moreover, we show that the category of generalized continuous posets and continuous morphisms is dually equivalent to the category of F-spaces and F-morphisms. In particular, some special cases are obtained, such as the topological representations for posets, domains, continuous lattices and join-semilattices.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139480389","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
Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality 非对称算子的兼容结构、马宁积和科斯祖尔对偶性
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2024-01-10 DOI: 10.1007/s10485-023-09760-x
Huhu Zhang, Xing Gao, Li Guo
{"title":"Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality","authors":"Huhu Zhang,&nbsp;Xing Gao,&nbsp;Li Guo","doi":"10.1007/s10485-023-09760-x","DOIUrl":"10.1007/s10485-023-09760-x","url":null,"abstract":"<div><p>Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking a uniform approach, this paper presents an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly stronger restraints among the replicated copies. The linear compatibility is in Koszul duality to the total compatibility, while the matching compatibility is self dual. Further, each compatibility condition can be expressed in terms of either one or both of the two Manin square products. Finally it is shown that the operads defined by these compatibility conditions from the associative algebra and differential algebra are Koszul utilizing rewriting systems.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139409568","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
Generalization of the Dehornoy–Lafont Order Complex to Categories: Application to Exceptional Braid Groups 德霍诺伊-拉丰阶复数在范畴中的泛化：特殊辫状群的应用
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-12-12 DOI: 10.1007/s10485-023-09757-6
Owen Garnier
{"title":"Generalization of the Dehornoy–Lafont Order Complex to Categories: Application to Exceptional Braid Groups","authors":"Owen Garnier","doi":"10.1007/s10485-023-09757-6","DOIUrl":"10.1007/s10485-023-09757-6","url":null,"abstract":"<div><p>The homology of a Garside monoid, thus of a Garside group, can be computed efficiently through the use of the order complex defined by Dehornoy and Lafont. We construct a categorical generalization of this complex and we give some computational techniques which are useful for reducing computing time. We then use this construction to complete results of Salvetti, Callegaro and Marin regarding the homology of exceptional complex braid groups. We most notably study the case of the Borchardt braid group <span>(B(G_{31}))</span> through its associated Garside category.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138581747","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
Koszul Monoids in Quasi-abelian Categories 准阿贝尔范畴中的科斯祖尔单体
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-12-06 DOI: 10.1007/s10485-023-09756-7
Rhiannon Savage
{"title":"Koszul Monoids in Quasi-abelian Categories","authors":"Rhiannon Savage","doi":"10.1007/s10485-023-09756-7","DOIUrl":"10.1007/s10485-023-09756-7","url":null,"abstract":"<div><p>Suppose that we have a bicomplete closed symmetric monoidal quasi-abelian category <span>(mathcal {E})</span> with enough flat projectives, such as the category of complete bornological spaces <span>({{textbf {CBorn}}}_k)</span> or the category of inductive limits of Banach spaces <span>({{textbf {IndBan}}}_k)</span>. Working with monoids in <span>(mathcal {E})</span>, we can generalise and extend the Koszul duality theory of Beilinson, Ginzburg, Soergel. We use an element-free approach to define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders’ embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of certain subcategories of the derived categories of graded modules over Koszul monoids and their duals.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09756-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547668","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 Sheaves on Generalised Spaces 广义空间上的同伦轴
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-12-04 DOI: 10.1007/s10485-023-09754-9
Severin Bunk
{"title":"Homotopy Sheaves on Generalised Spaces","authors":"Severin Bunk","doi":"10.1007/s10485-023-09754-9","DOIUrl":"10.1007/s10485-023-09754-9","url":null,"abstract":"<div><p>We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding preserves homotopy sheaves valued in (sufficiently nice) simplicial model categories. Moreover, we show that this induces an equivalence between sheaves of spaces on the original category and colimit-preserving sheaves of spaces on its free cocompletion. We present three applications in geometry and topology: first, we prove that diffeological vector bundles descend along subductions of diffeological spaces. Second, we deduce that various flavours of bundle gerbes with connection satisfy <span>((infty ,2))</span>-categorical descent. Finally, we investigate smooth diffeomorphism actions in smooth bordism-type field theories on a manifold. We show how these smooth actions allow us to extract the values of a field theory on any object coherently from its values on generating objects of the bordism category.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-023-09754-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138485110","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
Unbounded Algebraic Derivators 无界代数导数
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-12-02 DOI: 10.1007/s10485-023-09752-x
Leovigildo Alonso Tarrío, Beatriz Álvarez Díaz, Ana Jeremías López
{"title":"Unbounded Algebraic Derivators","authors":"Leovigildo Alonso Tarrío,&nbsp;Beatriz Álvarez Díaz,&nbsp;Ana Jeremías López","doi":"10.1007/s10485-023-09752-x","DOIUrl":"10.1007/s10485-023-09752-x","url":null,"abstract":"<div><p>We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138475583","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
Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories 一般组合模型范畴中经由同伦末端的同伦极限
IF 0.6 4区 数学
Applied Categorical Structures Pub Date : 2023-11-27 DOI: 10.1007/s10485-023-09747-8
Sergey Arkhipov, Sebastian Ørsted
{"title":"Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories","authors":"Sergey Arkhipov,&nbsp;Sebastian Ørsted","doi":"10.1007/s10485-023-09747-8","DOIUrl":"10.1007/s10485-023-09747-8","url":null,"abstract":"<div><p>We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield–Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138449161","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: 5
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