{"title":"Presentations of Pseudodistributive Laws","authors":"Charles Walker","doi":"10.1007/s10485-024-09798-5","DOIUrl":"10.1007/s10485-024-09798-5","url":null,"abstract":"<div><p>By considering the situation in which the involved pseudomonads are presented in no-iteration form, we deduce a number of alternative presentations of pseudodistributive laws including a “decagon” form, a pseudoalgebra form, a no-iteration form, and a warping form. As an application, we show that five coherence axioms suffice in the usual monoidal definition of a pseudodistributive law.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912800","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}
{"title":"Counting Functions for Random Objects in a Category","authors":"Brandon Alberts","doi":"10.1007/s10485-024-09797-6","DOIUrl":"10.1007/s10485-024-09797-6","url":null,"abstract":"<div><p>In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below <i>X</i> is studied as <span>(Xrightarrow infty )</span>. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure <span>(mu )</span> on the isomorphism classes of the category with sufficient respect for a product structure, we prove a version of the Law of Large Numbers to give the asymptotic growth rate as <i>X</i> tends towards <span>(infty )</span> of such functions with probability 1 in terms of the finite moments of <span>(mu )</span> and the ordering. Such counting functions are motivated by work in arithmetic statistics, including number field counting as in Malle’s conjecture and point counting as in the Batyrev–Manin conjecture. Recent work of Sawin–Wood gives sufficient conditions to construct such a measure <span>(mu )</span> from a well-behaved sequence of finite moments in very broad contexts, and we prove our results in this broad context with the added assumption that a product structure in the category is respected. These results allow us to formalize vast heuristic predictions about counting functions in general settings.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912801","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}
{"title":"Non-Abelian Extensions of Groupoids and Their Groupoid Rings","authors":"Natã Machado, Johan Öinert, Stefan Wagner","doi":"10.1007/s10485-024-09795-8","DOIUrl":"10.1007/s10485-024-09795-8","url":null,"abstract":"<div><p>We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids <span>({mathcal {N}}rightarrow {mathcal {E}}rightarrow {mathcal {G}})</span> gives rise to a groupoid crossed product of <span>({mathcal {G}})</span> by the groupoid ring of <span>({mathcal {N}})</span> which recovers the groupoid ring of <span>({mathcal {E}})</span> up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09795-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142826110","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}
G. S. H. Cruttwell, Jean-Simon Pacaud Lemay, Elias Vandenberg
{"title":"A Tangent Category Perspective on Connections in Algebraic Geometry","authors":"G. S. H. Cruttwell, Jean-Simon Pacaud Lemay, Elias Vandenberg","doi":"10.1007/s10485-024-09796-7","DOIUrl":"10.1007/s10485-024-09796-7","url":null,"abstract":"<div><p>There is an abstract notion of connection in any tangent category. In this paper, we show that when applied to the tangent category of affine schemes, this recreates the classical notion of a connection on a module (and similarly, in the tangent category of schemes, this recreates the notion of connection on a quasi-coherent sheaf of modules). By contrast, we also show that in the tangent category of algebras, there are no non-trivial connections.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142798557","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}
{"title":"Bi-accessible and Bipresentable 2-Categories","authors":"Ivan Di Liberti, Axel Osmond","doi":"10.1007/s10485-024-09794-9","DOIUrl":"10.1007/s10485-024-09794-9","url":null,"abstract":"<div><p>We develop a 2-dimensional version of accessibility and presentability compatible with the formalism of flat pseudofunctors. First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and cofinality; in particular we show that <span>(sigma )</span>-<i>filteredness</i> and <i>bifilteredness</i> are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of <i>bicompact</i> objects and <i>bifiltered</i> bicolimits. We then characterize them as categories of <i>flat pseudofunctors</i>. We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small <i>bilex</i> 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of finitary 2-monads on <span>(textbf{Cat})</span> are finitely bipresentable, which in particular captures the case of <span>(textbf{Lex})</span>, the 2-category of small lex categories. Invoking the technology of <i>lex-colimits</i>, we prove further that several 2-categories arising in categorical logic (<b>Reg, Ex, Coh, Ext, Adh, Pretop</b>) are also finitely bipresentable.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09794-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142798442","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}
{"title":"An Equivalence Between Two Models of (infty )-Categories of Enriched Presheaves","authors":"Hadrian Heine","doi":"10.1007/s10485-024-09792-x","DOIUrl":"10.1007/s10485-024-09792-x","url":null,"abstract":"<div><p>Let <span>({{mathcal {O}}}rightarrow {text {BM}})</span> be a <span>({text {BM}})</span>-operad that exhibits an <span>(infty )</span>-category <span>({{mathcal {D}}})</span> as weakly bitensored over non-symmetric <span>(infty )</span>-operads <span>({{mathcal {V}}}rightarrow text {Ass }, {{mathcal {W}}}rightarrow text {Ass })</span> and <span>({{mathcal {C}}})</span> a <span>({{mathcal {V}}})</span>-enriched <span>(infty )</span>-precategory. We construct an equivalence </p><div><div><span>$$begin{aligned} text {Fun}_{text {Hin}}^{{mathcal {V}}}({{mathcal {C}}},{{mathcal {D}}}) simeq text {Fun}^{{mathcal {V}}}({{mathcal {C}}},{{mathcal {D}}}) end{aligned}$$</span></div></div><p>of <span>(infty )</span>-categories weakly right tensored over <span>({{mathcal {W}}})</span> between Hinich’s construction of <span>({{mathcal {V}}})</span>-enriched functors of Hinich (Adv Math 367:107129, 2020) and our construction of <span>({{mathcal {V}}})</span>-enriched functors of Heine (Adv Math 417:108941, 2023).\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09792-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714185","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}
{"title":"Operad Structures on the Species Composition of Two Operads","authors":"Imen Rjaiba","doi":"10.1007/s10485-024-09793-w","DOIUrl":"10.1007/s10485-024-09793-w","url":null,"abstract":"<div><p>We give an explicit description of two operad structures on the species composition <span>(textbf{p}circ textbf{q})</span>, where <span>(textbf{q})</span> is any given positive operad, and where <span>(textbf{p})</span> is the <span>({text{ NAP } })</span> operad, or a shuffle version of the magmatic operad <span>({text{ Mag } })</span>. No distributive law between <span>(textbf{p})</span> and <span>(textbf{q})</span> is assumed.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142691875","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}
{"title":"Dualizations of Approximations, (aleph _1)-Projectivity, and Vopěnka’s Principles","authors":"Asmae Ben Yassine, Jan Trlifaj","doi":"10.1007/s10485-024-09791-y","DOIUrl":"10.1007/s10485-024-09791-y","url":null,"abstract":"<div><p>The approximation classes of modules that arise as components of cotorsion pairs are tied up by Salce’s duality. Here we consider general approximation classes of modules and investigate possibilities of dualization in dependence on closure properties of these classes. While some proofs are easily dualized, other dualizations require large cardinal principles, and some fail in ZFC, with counterexamples provided by classes of <span>(aleph _1)</span>-projective modules over non-perfect rings. For example, we show that the statement “each covering class of modules closed under homomorphic images is of the form <span>({mathrm{Gen,}}(M))</span> for a module <i>M</i>” is equivalent to Vopěnka’s Principle.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09791-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142565800","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}
{"title":"Generalized Multicategories: Change-of-Base, Embedding, and Descent","authors":"Rui Prezado, Fernando Lucatelli Nunes","doi":"10.1007/s10485-024-09775-y","DOIUrl":"10.1007/s10485-024-09775-y","url":null,"abstract":"<div><p>Via the adjunction <span>( - *mathbbm {1} dashv mathcal V(mathbbm {1},-) :textsf {Span}({mathcal {V}}) rightarrow {mathcal {V}} text {-} textsf {Mat} )</span> and a cartesian monad <i>T</i> on an extensive category <span>( {mathcal {V}} )</span> with finite limits, we construct an adjunction <span>( - *mathbbm {1} dashv {mathcal {V}}(mathbbm {1},-) :textsf {Cat}(T,{mathcal {V}}) rightarrow ({overline{T}}, mathcal V)text{- }textsf{Cat} )</span> between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad <i>T</i> satisfies a suitable property, which holds for several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor <span>( - *mathbbm {1} :textsf {Set} rightarrow {mathcal {V}} )</span> is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures. These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09775-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142555266","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}
{"title":"Partial Algebras and Implications of (Weak) Matrix Properties","authors":"Michael Hoefnagel, Pierre-Alain Jacqmin","doi":"10.1007/s10485-024-09790-z","DOIUrl":"10.1007/s10485-024-09790-z","url":null,"abstract":"<div><p>Matrix properties are a type of property of categories which includes the ones of being Mal’tsev, arithmetical, majority, unital, strongly unital, and subtractive. Recently, an algorithm has been developed to determine implications <span>(textrm{M}Rightarrow _{textrm{lex}_*}textrm{N})</span> between them. We show here that this algorithm reduces to constructing a partial term corresponding to <span>(textrm{N})</span> from a partial term corresponding to <span>(textrm{M})</span>. Moreover, we prove that this is further equivalent to the corresponding implication between the weak versions of these properties, i.e., the one where only strong monomorphisms are considered instead of all monomorphisms.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"32 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09790-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142519128","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}