On functors which are lax epimorphisms

Abstract

We show that lax epimorphisms in the category Cat are precisely the functors P : Ε → 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 colimit of all arrows P(E) → B for E in E. Secondly, lax epimorphisms are precisely the functors P such that for every morphism f of B the category of all factorizations through objects of P[E] is connected. A relationship between pseudoepimorphisms and lax epimorphisms is discussed.