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On Functors Which Are Lax Epimorphisms

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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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