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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)
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)
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.
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