On Functors Which Are Lax Epimorphisms

Jiri Adamek, Robert El Bashir, Manuela Sobral, Jiri Velebil

We show that lax epimorphisms in the category Cat are precisely the functors $P : {\cal E} \to {\cal B}$ for which the functor $P^{*}: [{\cal B}, Set] \to [{\cal E}, Set]$ 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 ${\cal B}$ is an absolute colimit of all arrows $P(E)\to B$ for $E$ in ${\cal E}$. Secondly, lax epimorphisms are precisely the functors $P$ such that for every morphism $f$ of ${\cal B}$ the category of all factorizations through objects of $P[{\cal E}]$ is connected.

A relationship between pseudoepimorphisms and lax epimorphisms is discussed.

Keywords: lax epimorphism.

2000 MSC: 18A20.

Theory and Applications of Categories, Vol. 8, 2001, No. 20, pp 509-521.


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