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Abstract: Deconstructibility is an often-used sufficient condition on a class $$\mathcal {C}$$ of modules that allows one to carry out homological algebra relative to $$\mathcal {C}$$. The principle Maximum Deconstructibility (MD) asserts that a certain necessary condition for a class to be deconstructible is also sufficient. MD implies, for example, that the classes of Gorenstein Projective modules, Ding Projective modules, their relativized variants, and all torsion classes are deconstructible over any ring. MD was known to follow from Vopěnka’s Principle and imply the existence of an $$\omega _1$$-strongly compact cardinal. We prove that MD is equivalent to Vopěnka’s Principle, and to the assertion that each torsion class of abelian groups is generated by a single group within the class (yielding the converse of a theorem of Göbel and Shelah). PubDate: 2025-05-31
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Abstract: In this paper, a notion of 0-ideals on a set is proposed and further it is shown that 0-ideals give rise to a power-enriched monad $$\mathbb {Z}$$=$$({\textbf{Z}},m,e)$$ on the category of sets, namely a 0-ideal monad. As a first application, a new characterization of approach spaces is given by verifying that the category $${\mathbb {Z}}$$-Mon of $${\mathbb {Z}}$$-monoids is isomorphic to the category App of approach spaces. The second application consists of two components: (i) Based on 0-ideals, a concept of approach 0-convergence spaces is introduced. (ii) By using the Kleisli extension of $${\textbf{Z}}$$, the existence of an isomorphism between the category AConv of approach 0-convergence spaces and the category $${(\mathbb {Z},2)}$$-Cat of relational $${\mathbb {Z}}$$-algebras is verified. Then from the fact that $${\mathbb {Z}}$$-Mon and $${(\mathbb {Z},2)}$$-Cat are isomorphic, another new description of approach spaces is obtained by an isomorphism between AConv and App. PubDate: 2025-05-31
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Abstract: We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and linear contractions in terms of simple category-theoretic structures and properties that do not refer to norms, continuity, or real numbers. This builds on Heunen, Kornell, and Van der Schaaf’s easier characterisation of the category of all Hilbert spaces and linear contractions. PubDate: 2025-05-26
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Abstract: In this paper, we present an operadic characterization of convexity utilizing a PROP which governs convex structures and derive several convex Grothendieck constructions. Our main focus is a Grothendieck construction which simultaneously captures convex structures and monoidal structures on categories. Our proof of this Grothendieck construction makes heavy use of both our operadic characterization of convexity and operads governing monoidal structures. We apply these new tools to two key concepts: entropy in information theory and quantum contextuality in quantum foundations. In the former, we explain that Baez, Fritz, and Leinster’s categorical characterization of entropy has a more natural formulation in terms of continuous, convex-monoidal functors out of convex Grothendieck constructions; and in the latter we show that certain convex monoids used to characterize contextual distributions naturally arise as convex Grothendieck constructions. PubDate: 2025-05-24
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Abstract: The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem. PubDate: 2025-05-16
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Abstract: Under the name of hardly groupoid, we investigate the (internal) categories in which any morphism is both monomorphic and epimorphic. It is the case for any internal category in a congruence modular variety. In the more general context of Gumm categories, we explore the large diversity of situation determined by this notion. PubDate: 2025-04-05
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Abstract: In this paper we study a generalization of the notion of AS-regularity for connected $${\mathbb Z}$$-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 $${\mathbb Z}$$-algebras, which we call quantum projective $${\mathbb Z}$$-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 $${\mathbb Z}$$-algebras as homogeneous coordinate algebras. In particular, the latter are thus noncommutative $${\mathbb P}^1\times {\mathbb P}^1$$ [in the sense of Van den Bergh (Int Math Res Not 17:3983–4026, 2011)]. PubDate: 2025-03-29
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Abstract: 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. PubDate: 2025-03-22
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Abstract: 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 $$\textbf{Set}$$-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as "Sh(B)-valued models"). For the coherent fragment $$L_{\omega \omega }^g \subseteq L_{\omega \omega }$$ this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to $$L_{\kappa \kappa }^g$$ when $$\kappa $$ is weakly compact. We present some further applications: first, a Sh(B)-valued completeness theorem for $$L_{\kappa \kappa }^g$$ ($$\kappa $$ is weakly compact), second, that $$\mathcal {C}\rightarrow \textbf{Set} $$ regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors. PubDate: 2025-03-11
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Abstract: 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. PubDate: 2025-03-06
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Abstract: We study restrictions of the correspondence between the lattice $$\textsf{SE}(L)$$ of strongly exact filters, of a frame L, and the coframe $$\mathcal {S}_o(L)$$ of fitted sublocales. In particular, we consider the classes of exact filters $$\textsf{E}(L)$$, regular filters $$\textsf{R}(L)$$, and the intersections $$\mathcal {J}(\textsf{CP}(L))$$ and $$\mathcal {J}(\textsf{SO}(L))$$ of completely prime and Scott-open filters, respectively. We show that all these classes of filters are sublocales of $$\textsf{SE}(L)$$ and as such correspond to subcolocales of $$\mathcal {S}_o(L)$$ 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. PubDate: 2025-02-28
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Abstract: For a plural signature $$\Sigma $$ and with regard to the category $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$, of naturally preordered idempotent $$\Sigma $$-algebras and surjective homomorphisms, we define a contravariant functor $$\textrm{Lsys}_{\Sigma }$$ from $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ to $$\textsf {Cat}$$, the category of categories, that assigns to $${\textbf {I}}$$ in $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ the category $${\textbf {I}}$$-$$\textsf {LAlg}(\Sigma )$$, of $${\textbf {I}}$$-semi-inductive Lallement systems of $$\Sigma $$-algebras, and a covariant functor $$(\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, \cdot )$$ from $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ to $$\textsf {Cat}$$, that assigns to $${\textbf {I}}$$ in $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ the category $$(\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, {\textbf {I}})$$, of the coverings of $${\textbf {I}}$$, i.e., the ordered pairs $$({\textbf {A}},f)$$ in which $${\textbf {A}}$$ is a $$\Sigma $$-algebra and a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories $$\int ^{\textsf {NPIAlg}(\Sigma )_{\textsf {s}}}\textrm{Lsys}_{\Sigma }$$ and $$\int _{\textsf {NPIAlg}(\Sigma )_{\textsf {s}}}(\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, \cdot )$$; define a functor $$\mathfrak {L}_{\Sigma }$$ 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. PubDate: 2025-02-08
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Abstract: The present article aims to develop a categorical duality for the category of bounded complete J-algebraic lattices. In terms of the lattice of weak ideals, we first construct a left adjoint to the forgetful functor Sup$$\rightarrow $$ $${\textbf {Pos}}_\vee $$, where Sup is the category of complete lattices and join-preserving maps and $${\textbf {Pos}}_\vee $$ is the category of posets and maps that preserve existing binary joins. Based on which, we propose the concept of W-structures over posets and give a W-structure representation for bounded complete J-algebraic posets, which generalizes the representation of algebraic lattices. Finally, we show that the category of join-semilattice WS-structures and homomorphisms is dually equivalent to the category of bounded complete J-algebraic lattices and homomorphisms. PubDate: 2025-02-04
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Abstract: 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. PubDate: 2025-01-03
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Abstract: In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below X is studied as $$X\rightarrow \infty $$. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure $$\mu $$ 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 X tends towards $$\infty $$ of such functions with probability 1 in terms of the finite moments of $$\mu $$ 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 $$\mu $$ 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. PubDate: 2025-01-03
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Abstract: 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 $${\mathcal {N}}\rightarrow {\mathcal {E}}\rightarrow {\mathcal {G}}$$ gives rise to a groupoid crossed product of $${\mathcal {G}}$$ by the groupoid ring of $${\mathcal {N}}$$ which recovers the groupoid ring of $${\mathcal {E}}$$ 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. PubDate: 2024-12-16
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Abstract: 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. PubDate: 2024-12-11
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Abstract: 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 $$\sigma $$-filteredness and bifilteredness are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of bicompact objects and bifiltered bicolimits. We then characterize them as categories of flat pseudofunctors. We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small bilex 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of finitary 2-monads on $$\textbf{Cat}$$ are finitely bipresentable, which in particular captures the case of $$\textbf{Lex}$$, the 2-category of small lex categories. Invoking the technology of lex-colimits, we prove further that several 2-categories arising in categorical logic (Reg, Ex, Coh, Ext, Adh, Pretop) are also finitely bipresentable. PubDate: 2024-12-09
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Abstract: Let $${{\mathcal {O}}}\rightarrow {\text {BM}}$$ be a $${\text {BM}}$$-operad that exhibits an $$\infty $$-category $${{\mathcal {D}}}$$ as weakly bitensored over non-symmetric $$\infty $$-operads $${{\mathcal {V}}}\rightarrow \text {Ass }, {{\mathcal {W}}}\rightarrow \text {Ass }$$ and $${{\mathcal {C}}}$$ a $${{\mathcal {V}}}$$-enriched $$\infty $$-precategory. We construct an equivalence $$\begin{aligned} \text {Fun}_{\text {Hin}}^{{\mathcal {V}}}({{\mathcal {C}}},{{\mathcal {D}}}) \simeq \text {Fun}^{{\mathcal {V}}}({{\mathcal {C}}},{{\mathcal {D}}}) \end{aligned}$$ Fun Hin V ( C , D ) ≃ Fun V ( C , D ) of $$\infty $$-categories weakly right tensored over $${{\mathcal {W}}}$$ between Hinich’s construction of $${{\mathcal {V}}}$$-enriched functors of Hinich (Adv Math 367:107129, 2020) and our construction of $${{\mathcal {V}}}$$-enriched functors of Heine (Adv Math 417:108941, 2023). PubDate: 2024-11-26
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Abstract: We give an explicit description of two operad structures on the species composition $$\textbf{p}\circ \textbf{q}$$, where $$\textbf{q}$$ is any given positive operad, and where $$\textbf{p}$$ is the $${\text{ NAP } }$$ operad, or a shuffle version of the magmatic operad $${\text{ Mag } }$$. No distributive law between $$\textbf{p}$$ and $$\textbf{q}$$ is assumed. PubDate: 2024-11-22
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