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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.
@article{TAC_2001_8_a19,
author = {Jiri Adamek and Robert El Bashir and Manuela Sobral and Jiri Velebil},
title = {On {Functors} {Which} {Are} {Lax} {Epimorphisms}},
journal = {Theory and applications of categories},
pages = {509--521},
publisher = {mathdoc},
volume = {8},
year = {2001},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2001_8_a19/}
}
TY - JOUR AU - Jiri Adamek AU - Robert El Bashir AU - Manuela Sobral AU - Jiri Velebil TI - On Functors Which Are Lax Epimorphisms JO - Theory and applications of categories PY - 2001 SP - 509 EP - 521 VL - 8 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TAC_2001_8_a19/ LA - en ID - TAC_2001_8_a19 ER -
Jiri Adamek; Robert El Bashir; Manuela Sobral; Jiri Velebil. On Functors Which Are Lax Epimorphisms. Theory and applications of categories, Tome 8 (2001), pp. 509-521. http://geodesic.mathdoc.fr/item/TAC_2001_8_a19/