Applied Categorical Structures最新文献

We introduce intermediate commutators and study their degrees. We define ((q, {}))-capable groups and prove that a group G is ((q, {}))-capable if and only if (Z^{wedge }_{(q, {})}(G)=1).

Let ({mathbb {A}}) be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the two-dimensional cokernel diagram of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category ({mathbb {A}}).

In this paper, we provide a notion of (infty )-bicategories fibred in (infty )-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling (infty )-bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an (infty )-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of ({text {Set}}^+_{Delta })-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories.

In a recent paper (Hoefnagel et al. in Theory Appl Categ 38:737–790, 2022), an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices—the so called ‘matrix properties’. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal’tsev, majority and arithmetical categories. Using a computer implementation of our algorithm, we can display all such properties given by matrices of fixed dimensions, grouped according to their Bourn localizations, as well as the implications between them.

In ordinary category theory, limits are known to be equivalent to terminal objects in the slice category of cones. In this paper, we prove that the 2-categorical analogues of this theorem relating 2-limits and 2-terminal objects in the various choices of slice 2-categories of 2-cones are false. Furthermore we show that, even when weakening the 2-cones to pseudo- or lax-natural transformations, or considering bi-type limits and bi-terminal objects, there is still no such correspondence.

In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in S. Celani, L.J. González (Appl Categ Struct 28:853–875, 2020) for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies.

This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.

We consider the ordinary category (mathsf {Span}({mathcal {C}})) of (isomorphism classes of) spans of morphisms in a category (mathcal {C}) with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of (mathsf {Span}({mathcal {C}})) to be an allegory. In particular, when ({mathcal {C}}) carries a pullback-stable, but not necessarily proper, (({mathcal {E}},{mathcal {M}}))-factorization system, we establish a quotient category (mathsf {Span}_{{mathcal {E}}}({mathcal {C}})) that is isomorphic to the category (mathsf {Rel}_{{mathcal {M}}}({mathcal {C}})) of ({mathcal {M}})-relations in ({mathcal {C}}), and show that it is a (unitary and tabular) allegory precisely when ({mathcal {M}}) is a class of monomorphisms in ({mathcal {C}}). Without the restriction to monomorphisms, one can still find a least pullback-stable and composition-closed class ({mathcal {E}}_{bullet }) containing (mathcal E) such that (mathsf {Span}_{{mathcal {E}}_{bullet }}({mathcal {C}})) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the 2-category of all regular categories.

In a regular category (mathbb {E}), the direct image along a regular epimorphism f of a preorder is not a preorder in general. In Set, its best preorder approximation is then its cocartesian image above f. In a regular category, the existence of such a cocartesian image above f of a preorder S is actually equivalent to the existence of the supremum (R[f]vee S) among the preorders. We investigate here some conditions ensuring the existence of these cocartesian images or equivalently of these suprema. They apply to two very dissimilar contexts: any topos (mathbb {E}) with suprema of countable chains of subobjects or any n-permutable regular category.