Applied Categorical Structures最新文献

We prove that for any bisimplicial set X, the natural comparison map between the diagonal dX and the total simplicial set TX is a categorical equivalence in the sense of Joyal and Lurie.

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.

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).

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.

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.

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.

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.

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}^1times {mathbb P}^1) [in the sense of Van den Bergh (Int Math Res Not 17:3983–4026, 2011)].

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.

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.