Applied Categorical Structures最新文献

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.

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.

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 (Xrightarrow 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.

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.

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.

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.

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

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).

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.

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 (aleph _1)-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 ({mathrm{Gen,}}(M)) for a module M” is equivalent to Vopěnka’s Principle.

Via the adjunction ( - *mathbbm {1} dashv mathcal V(mathbbm {1},-) :textsf {Span}({mathcal {V}}) rightarrow {mathcal {V}} text {-} textsf {Mat} ) and a cartesian monad T on an extensive category ( {mathcal {V}} ) with finite limits, we construct an adjunction ( - *mathbbm {1} dashv {mathcal {V}}(mathbbm {1},-) :textsf {Cat}(T,{mathcal {V}}) rightarrow ({overline{T}}, mathcal V)text{- }textsf{Cat} ) between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad T 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 ( - *mathbbm {1} :textsf {Set} rightarrow {mathcal {V}} ) 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.