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Applied Categorical Structures最新文献

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On Internal Categories and Crossed Objects in the Category of Monoids 论一元群范畴中的内范畴和交叉对象
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-08-14 DOI: 10.1007/s10485-025-09822-2
Ilia Pirashvili
{"title":"On Internal Categories and Crossed Objects in the Category of Monoids","authors":"Ilia Pirashvili","doi":"10.1007/s10485-025-09822-2","DOIUrl":"10.1007/s10485-025-09822-2","url":null,"abstract":"<div><p>In a previous work on quadratic algebras (Pirashvili in Glas Math J 61: 151–167, 2018), I constructed an internal category in the category of monoids, recalled in Sect. 3.2.1. Based on this, we introduce the notion of a crossed semi-bimodule in this paper. This new construction generalises the notion of a crossed semi-module, introduced independently by R. Street and A. Patchkoria, see Joyal (Macquarie Math Reports 860081, 1986) and Patchkoria (Georg Math J 5: 575–581, 1986) respectively. We also show that there is a one to one correspondence between crossed semi-bimodules and strict monoidal category structures on transformation categories satisfying the cc-condition, see Sects. 4 and 5.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144832120","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
The Leibniz PROP is a Crossed Presimplicial Algebra 莱布尼茨PROP是一个交叉预简单代数
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-07-30 DOI: 10.1007/s10485-025-09820-4
Murat Can Aşkaroğulları, Atabey Kaygun
{"title":"The Leibniz PROP is a Crossed Presimplicial Algebra","authors":"Murat Can Aşkaroğulları,&nbsp;Atabey Kaygun","doi":"10.1007/s10485-025-09820-4","DOIUrl":"10.1007/s10485-025-09820-4","url":null,"abstract":"<div><p>We prove that the Leibniz PROP is isomorphic as <span>(Bbbk )</span>-linear categories (not as monoidal categories) to the symmetric crossed presimplicial algebra <span>(Bbbk [(Delta ^+)^{op} mathbb {S}])</span> where <span>(Delta ^+)</span> is the skeletal category of finite well-ordered sets with surjections, but the distributive law between <span>((Delta ^+)^{op})</span> and the symmetric groups <span>(mathbb {S} = bigsqcup _{nge 1} S_n)</span> is not the standard one.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145171785","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
Monoidal Envelopes of Families of (infty )-Operads and (infty )-Operadic Kan Extensions (infty ) -操作数和(infty ) -操作数Kan扩展族的一元包络
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-07-18 DOI: 10.1007/s10485-025-09821-3
Kensuke Arakawa
{"title":"Monoidal Envelopes of Families of (infty )-Operads and (infty )-Operadic Kan Extensions","authors":"Kensuke Arakawa","doi":"10.1007/s10485-025-09821-3","DOIUrl":"10.1007/s10485-025-09821-3","url":null,"abstract":"<div><p>We provide details of the proof of Lurie’s theorem on operadic Kan extensions. Along the way, we generalize the construction of monoidal envelopes of <span>(infty )</span>-operads to families of <span>(infty )</span>-operads and use it to construct the fiberwise direct sum functor, both of which we characterize by certain universal properties. Aside from their use in elaborating the proof of Lurie’s theorem, these results and constructions have their independent interest.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145166797","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
Double Groupoids and 2-Groupoids in Regular Mal’tsev Categories 正则马尔采夫范畴中的双群拟和2群拟
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-07-16 DOI: 10.1007/s10485-025-09819-x
Nadja Egner, Marino Gran
{"title":"Double Groupoids and 2-Groupoids in Regular Mal’tsev Categories","authors":"Nadja Egner,&nbsp;Marino Gran","doi":"10.1007/s10485-025-09819-x","DOIUrl":"10.1007/s10485-025-09819-x","url":null,"abstract":"<div><p>We prove that the category 2-<span>(textrm{Grpd}(mathscr {C}))</span> of internal 2-groupoids is a Birkhoff subcategory of the category <span>(textrm{Grpd}^2(mathscr {C}))</span> of double groupoids in a regular Mal’tsev category <span>(mathscr {C})</span> with finite colimits, and we provide a simple description of the reflector. In particular, when <span>(mathscr {C})</span> is a Mal’tsev variety of universal algebras, the category 2-<span>(textrm{Grpd}(mathscr {C}))</span> is also a Mal’tsev variety, of which we describe the corresponding algebraic theory. When <span>(mathscr {C})</span> is a naturally Mal’tsev category, the reflector from <span>(textrm{Grpd}^2(mathscr {C}))</span> to 2-<span>(textrm{Grpd}(mathscr {C}))</span> has an additional property related to the commutator of equivalence relations. We prove that the category 2-<span>(textrm{Grpd}(mathscr {C}))</span> is semi-abelian when <span>(mathscr {C})</span> is semi-abelian, and then provide sufficient conditions for 2-<span>(textrm{Grpd}(mathscr {C}))</span> to be action representable.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145165515","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
Monadic Aspects of the Ideal Lattice Functor on the Category of Distributive Lattices 分配格范畴上理想格函子的一元方面
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-07-10 DOI: 10.1007/s10485-025-09811-5
Ando Razafindrakoto
{"title":"Monadic Aspects of the Ideal Lattice Functor on the Category of Distributive Lattices","authors":"Ando Razafindrakoto","doi":"10.1007/s10485-025-09811-5","DOIUrl":"10.1007/s10485-025-09811-5","url":null,"abstract":"<div><p>It is known that the construction of the frame of ideals from a distributive lattice induces a monad whose algebras are precisely the frames and frame homomorphisms. Using the Fakir construction of an idempotent approximation of a monad, we extend B. Jacobs’ results on lax idempotent monads and show that the sequence of monads and comonads generated by successive iterations of this ideal functor on its algebras and coalgebras do not strictly lead to a new category. We further extend this result and provide a new proof of the equivalence between distributive lattices and coherent frames by showing that when the first inductive step in the Fakir construction is the identity monad, then the ambient category is equivalent to the category of free algebras.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09811-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145163615","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
Higher Catoids, Higher Quantales and their Correspondences 高等类元、高等量子及其对应关系
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-06-27 DOI: 10.1007/s10485-025-09817-z
Cameron Calk, Philippe Malbos, Damien Pous, Georg Struth
{"title":"Higher Catoids, Higher Quantales and their Correspondences","authors":"Cameron Calk,&nbsp;Philippe Malbos,&nbsp;Damien Pous,&nbsp;Georg Struth","doi":"10.1007/s10485-025-09817-z","DOIUrl":"10.1007/s10485-025-09817-z","url":null,"abstract":"<div><p>We introduce <span>(omega )</span>-catoids as generalisations of (strict) <span>(omega )</span>-categories and in particular the higher path categories generated by computads or polygraphs in higher-dimensional rewriting. We also introduce <span>(omega )</span>-quantales that generalise the <span>(omega )</span>-Kleene algebras recently proposed for algebraic coherence proofs in higher-dimensional rewriting. We then establish correspondences between <span>(omega )</span>-catoids and convolution <span>(omega )</span>-quantales. These are related to Jónsson-Tarski-style dualisms between relational structures and lattices with operators. We extend these correspondences to <span>((omega,p))</span>-catoids, catoids with a groupoid structure above some dimension, and convolution <span>((omega,p))</span>-quantales, using Dedekind quantales above some dimension to capture homotopic constructions and proofs in higher-dimensional rewriting. We also specialise them to finitely decomposable <span>((omega, p))</span>-catoids, an appropriate setting for defining <span>((omega, p))</span>-semirings and <span>((omega, p))</span>-Kleene algebras. These constructions support the systematic development and justification of <span>(omega )</span>-Kleene algebra and <span>(omega )</span>-quantale axioms, improving on the recent approach mentioned, where axioms for <span>(omega )</span>-Kleene algebras have been introduced in an ad hoc fashion.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09817-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145170336","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
Choice-Free Duality for Tarski Algebras Tarski代数的无选择对偶性
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-06-25 DOI: 10.1007/s10485-025-09816-0
Sergio Arturo Celani, Luciano Javier González
{"title":"Choice-Free Duality for Tarski Algebras","authors":"Sergio Arturo Celani,&nbsp;Luciano Javier González","doi":"10.1007/s10485-025-09816-0","DOIUrl":"10.1007/s10485-025-09816-0","url":null,"abstract":"<div><p>In [N. Bezhanishvili and W. H. Holliday. Choice-free Stone duality. J. Symb. Log., 85(1):109–148, 2020.], the authors develop a choice-free topological duality for the algebraic category of Boolean algebras. We adapt the techniques and constructions given by Bezhanishvili and Holliday to develop a topological duality for the algebraic category of Tarski algebras without using the Axiom of Choice. Then, we show that the duality presented here for Tarski algebras is in fact a generalization of the duality given by Bezhanishvili and Holliday for Boolean algebras. We also obtain a choice-free topological duality for the algebraic category of generalized Boolean algebra.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145169974","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
Enriched Kleisli Objects for Pseudomonads 伪单胞菌富集Kleisli对象
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-06-16 DOI: 10.1007/s10485-025-09810-6
Adrian Miranda
{"title":"Enriched Kleisli Objects for Pseudomonads","authors":"Adrian Miranda","doi":"10.1007/s10485-025-09810-6","DOIUrl":"10.1007/s10485-025-09810-6","url":null,"abstract":"<div><p>A pseudomonad on a 2-category whose underlying endomorphism is a 2-functor can be seen as a diagram <span>(textbf{Psmnd} rightarrow textbf{Gray})</span> for which weighted limits and colimits can be considered. The 2-category of pseudoalgebras, pseudomorphisms and 2-cells is such a Gray-enriched weighted limit [21], however neither the Kleisli bicategory nor the 2-category of free pseudoalgebras are the analogous weighted colimit [13]. In this paper we describe the actual weighted colimit via a presentation, and show that the comparison 2-functor induced by any other pseudoadjunction splitting the original pseudomonad is bi-fully faithful. As a consequence, we see that biessential surjectivity on objects characterises left pseudoadjoints whose codomains have an ‘up to biequivalence’ version of the universal property for Kleisli objects. This motivates a homotopical study of Kleisli objects for pseudomonads, and to this end we show that the weight for Kleisli objects is cofibrant in the projective model structure on the enriched functor category <span>([textbf{Psmnd}^text {op} , textbf{Gray}])</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09810-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145166674","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
Discrete Vertices in Simplicial Objects Internal to a Monoidal Category 一元范畴内的简单对象中的离散顶点
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-06-11 DOI: 10.1007/s10485-025-09812-4
Arne Mertens
{"title":"Discrete Vertices in Simplicial Objects Internal to a Monoidal Category","authors":"Arne Mertens","doi":"10.1007/s10485-025-09812-4","DOIUrl":"10.1007/s10485-025-09812-4","url":null,"abstract":"<div><p>We follow the work of Aguiar (Internal categories and quantum groups. PhD Thesis, Cornell University, 1997) on internal categories and introduce simplicial objects internal to a monoidal category as certain colax monoidal functors. Then we compare three approaches to equipping them with a discrete set of vertices. We introduce based colax monoidal functors and show that under suitable conditions they are equivalent to the templicial objects defined by Lowen and Mertens (Algebr Geom Topol, 2024). We also compare templicial objects to the enriched Segal precategories appearing in Lurie ((Infinity,2)-categories and the Goodwillie calculus I. Preprint at https://arxiv.org/abs/0905.0462v2), Simpson (Homotopy theory of higher categories. New mathematical monographs, Cambridge University Press, Cambridge, vol 19, p 634, 2012, https://doi.org/10.1017/CBO9780511978111) and Bacard (Theory Appl Categ 35:1227–1267, 2020), and show that they are equivalent for cartesian monoidal categories, but not in general.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145165171","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 remark on the total simplicial set functor 关于全简单集函子的注解
IF 0.5 4区 数学
Applied Categorical Structures Pub Date : 2025-06-09 DOI: 10.1007/s10485-025-09815-1
Danny Stevenson
{"title":"A remark on the total simplicial set functor","authors":"Danny Stevenson","doi":"10.1007/s10485-025-09815-1","DOIUrl":"10.1007/s10485-025-09815-1","url":null,"abstract":"<div><p>We prove that for any bisimplicial set <i>X</i>, the natural comparison map between the diagonal <i>dX</i> and the total simplicial set <i>TX</i> is a categorical equivalence in the sense of Joyal and Lurie.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 3","pages":""},"PeriodicalIF":0.5,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09815-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145163866","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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