{"title":"A Categorical Characterization of Quantum Projective (mathbb {Z})-spaces","authors":"Izuru Mori, Adam Nyman","doi":"10.1007/s10485-025-09806-2","DOIUrl":"10.1007/s10485-025-09806-2","url":null,"abstract":"<div><p>In this paper we study a generalization of the notion of AS-regularity for connected <span>({mathbb Z})</span>-algebras defined in Mori and Nyman (J Pure Appl Algebra, 225(9), 106676, 2021). Our main result is a characterization of those categories equivalent to noncommutative projective schemes associated to right coherent regular <span>({mathbb Z})</span>-algebras, which we call quantum projective <span>({mathbb Z})</span>-spaces in this paper. As an application, we show that smooth quadric hypersurfaces and the standard noncommutative smooth quadric surfaces studied in Smith and Van den Bergh (J Noncommut Geom 7(3), 817–856, 2013) , Mori and Ueyama (J Noncommut Geom, 15(2), 489–529, 2021) have right noetherian AS-regular <span>({mathbb Z})</span>-algebras as homogeneous coordinate algebras. In particular, the latter are thus noncommutative <span>({mathbb P}^1times {mathbb P}^1)</span> [in the sense of Van den Bergh (Int Math Res Not 17:3983–4026, 2011)].</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143735430","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":"A Characterisation for the Category of Hilbert Spaces","authors":"Stephen Lack, Shay Tobin","doi":"10.1007/s10485-025-09805-3","DOIUrl":"10.1007/s10485-025-09805-3","url":null,"abstract":"<div><p>The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Solèr’s theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quarternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quarternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09805-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143667977","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":"Sh(B)-Valued Models of ((kappa ,kappa ))-Coherent Categories","authors":"Kristóf Kanalas","doi":"10.1007/s10485-025-09804-4","DOIUrl":"10.1007/s10485-025-09804-4","url":null,"abstract":"<div><p>A basic technique in model theory is to name the elements of a model by introducing new constant symbols. We describe the analogous construction in the language of syntactic categories/sites. As an application we identify <span>(textbf{Set})</span>-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as \"<i>Sh</i>(<i>B</i>)-valued models\"). For the coherent fragment <span>(L_{omega omega }^g subseteq L_{omega omega })</span> this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to <span>(L_{kappa kappa }^g)</span> when <span>(kappa )</span> is weakly compact. We present some further applications: first, a <i>Sh</i>(<i>B</i>)-valued completeness theorem for <span>(L_{kappa kappa }^g)</span> (<span>(kappa )</span> is weakly compact), second, that <span>(mathcal {C}rightarrow textbf{Set} )</span> regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09804-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594659","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":"Double Categories of Relations Relative to Factorisation Systems","authors":"Keisuke Hoshino, Hayato Nasu","doi":"10.1007/s10485-025-09799-y","DOIUrl":"10.1007/s10485-025-09799-y","url":null,"abstract":"<div><p>We relativise double categories of relations to stable orthogonal factorisation systems. Furthermore, we present the characterisation of the relative double categories of relations in two ways. The first utilises a generalised comprehension scheme, and the second focuses on a specific class of vertical arrows defined solely double-categorically. We organise diverse classes of double categories of relations and correlate them with significant classes of factorisation systems. Our framework embraces double categories of spans and double categories of relations on regular categories, which we meticulously compare to existing work on the characterisations of bicategories and double categories of spans and relations.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143564445","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":"Canonical extensions via fitted sublocales","authors":"Tomáš Jakl, Anna Laura Suarez","doi":"10.1007/s10485-025-09802-6","DOIUrl":"10.1007/s10485-025-09802-6","url":null,"abstract":"<div><p>We study restrictions of the correspondence between the lattice <span>(textsf{SE}(L))</span> of strongly exact filters, of a frame <i>L</i>, and the coframe <span>(mathcal {S}_o(L))</span> of fitted sublocales. In particular, we consider the classes of exact filters <span>(textsf{E}(L))</span>, regular filters <span>(textsf{R}(L))</span>, and the intersections <span>(mathcal {J}(textsf{CP}(L)))</span> and <span>(mathcal {J}(textsf{SO}(L)))</span> of completely prime and Scott-open filters, respectively. We show that all these classes of filters are sublocales of <span>(textsf{SE}(L))</span> and as such correspond to subcolocales of <span>(mathcal {S}_o(L))</span> with a concise description. The theory of polarities of Birkhoff is central to our investigations. We automatically derive universal properties for the said classes of filters by giving their descriptions in terms of polarities. The obtained universal properties strongly resemble that of the canonical extensions of lattices. We also give new equivalent definitions of subfitness in terms of the lattice of filters.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09802-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143513450","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":"Lallement Functor is a Weak Right Multiadjoint","authors":"J. Climent Vidal, E. Cosme Llópez","doi":"10.1007/s10485-025-09800-8","DOIUrl":"10.1007/s10485-025-09800-8","url":null,"abstract":"<div><p>For a plural signature <span>(Sigma )</span> and with regard to the category <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span>, of naturally preordered idempotent <span>(Sigma )</span>-algebras and surjective homomorphisms, we define a contravariant functor <span>(textrm{Lsys}_{Sigma })</span> from <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> to <span>(textsf {Cat})</span>, the category of categories, that assigns to <span>({textbf {I}})</span> in <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> the category <span>({textbf {I}})</span>-<span>(textsf {LAlg}(Sigma ))</span>, of <span>({textbf {I}})</span>-semi-inductive Lallement systems of <span>(Sigma )</span>-algebras, and a covariant functor <span>((textsf {Alg}(Sigma ),{downarrow _{textsf {s}}}, cdot ))</span> from <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> to <span>(textsf {Cat})</span>, that assigns to <span>({textbf {I}})</span> in <span>(textsf {NPIAlg}(Sigma )_{textsf {s}})</span> the category <span>((textsf {Alg}(Sigma ),{downarrow _{textsf {s}}}, {textbf {I}}))</span>, of the coverings of <span>({textbf {I}})</span>, i.e., the ordered pairs <span>(({textbf {A}},f))</span> in which <span>({textbf {A}})</span> is a <span>(Sigma )</span>-algebra and <img> a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories <span>(int ^{textsf {NPIAlg}(Sigma )_{textsf {s}}}textrm{Lsys}_{Sigma })</span> and <span>(int _{textsf {NPIAlg}(Sigma )_{textsf {s}}}(textsf {Alg}(Sigma ),{downarrow _{textsf {s}}}, cdot ))</span>; define a functor <span>(mathfrak {L}_{Sigma })</span> from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.\u0000</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-025-09800-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143370031","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":"Bounded complete J-algebraic lattices","authors":"Shengwei Han, Yu Xue","doi":"10.1007/s10485-025-09801-7","DOIUrl":"10.1007/s10485-025-09801-7","url":null,"abstract":"<div><p>The present article aims to develop a categorical duality for the category of bounded complete <i>J</i>-algebraic lattices. In terms of the lattice of weak ideals, we first construct a left adjoint to the forgetful functor <b>Sup</b><span>(rightarrow )</span> <span>({textbf {Pos}}_vee )</span>, where <b>Sup</b> is the category of complete lattices and join-preserving maps and <span>({textbf {Pos}}_vee )</span> is the category of posets and maps that preserve existing binary joins. Based on which, we propose the concept of <i>W</i>-structures over posets and give a <i>W</i>-structure representation for bounded complete <i>J</i>-algebraic posets, which generalizes the representation of algebraic lattices. Finally, we show that the category of join-semilattice <i>WS</i>-structures and homomorphisms is dually equivalent to the category of bounded complete <i>J</i>-algebraic lattices and homomorphisms.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143107886","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":"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}
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