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We show that lax epimorphisms in the category Cat are precisely the functors P : E −→ B for which the functor P * : [B, Set] −→ [E, Set] of composition with P is fully faithful. We present two other characterizations. Firstly, lax epimorphisms are precisely the " absolutely dense " functors, i.e., functors P such that every object B of B is an absolute(More)
Implications in a category can be presented as epimorphisms: an object satisfies the implication iff it is injective w.r.t. that epimorphism. G. Roçu formulated a logic for deriving an implication from other implications. We present two versions of im-plicational logics: a general one and a finitary one (for epimorphisms with finitely presentable domains(More)
A characterization of descent morphism in the category of Priestley spaces, as well as necessary and sufficient conditions for such morphisms to be effective are given. For that we embed this category in suitable categories of preordered topological spaces were descent and effective morphisms are described using the monadic description of descent. A(More)
For a C-indexed category A , an A-descent equivalence is a morphism of bundles in C which induces an equivalence between the A-descent categories of its domain and codomain. In this note, properties of such morphisms are studied, and those morphisms which are A-descent equivalences for all C-indexed categories A are fully characterized.
The categorical definition of semidirect products was introduced by D. Bourn and G. Janelidze in [2], where they proved that, in the category of groups, this notion coincides with the classical one. A characterization of pointed categories with categorical semidirect products was given in [3]. The existence of such products imply, in particular, that the(More)
We characterize the (effective) E-descent morphisms in the category Cat of small categories, when E is the class of discrete fibrations or the one of discrete cofibrations, and prove that every effective global-descent morphism is an effective E-descent morphism while its converse fails.
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