#
Finite categories with pushouts

##
D. Tambara

Let C be a finite category. For an object X of C one has the hom-functor
Hom(-,X) of C to Set. If G is a subgroup of Aut(X), one has the
quotient functor Hom(-,X)/G. We show that any finite product of hom-functors
of C is a sum of hom-functors if and only if C has pushouts and
coequalizers and that any finite product of hom-functors of C is a sum of
functors of the form \Hom(-,X)/G if and only if C has pushouts. These are
variations of the fact that a finite category has products if and only if it
has coproducts.

Keywords:
pushout, coequalizer, hom-functor, familially representable functor, nearly
representable functor

2010 MSC:
18A30, 18A35

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 30, pp 1017-1031.

Published 2015-07-27.

http://www.tac.mta.ca/tac/volumes/30/30/30-30.pdf

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