Applied Categorical Structures最新文献

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.

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.

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.

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).