 
Summary: ON FUNCTORS WHICH ARE LAX EPIMORPHISMS
JI
R AD
AMEK, ROBERT EL BASHIR, MANUELA SOBRAL, JI
R VELEBIL
ABSTRACT. 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, they 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.
1. Introduction
What are the epimorphisms of Cat, the category of small categories and functors? No
simple answer is known, and the present paper indicates that this may be a \wrong ques
tion", disregarding the 2categorical character of Cat. Anyway, with strong epimorphisms
we have more luck: as proved in [2], they are precisely those functors P : E ! B such
that every morphism of B is a composite of morphisms in P [E].
Our paper is devoted to lax epimorphisms in the 2category Cat. We follow the
