Let $\cal K$ be a locally finitely presentable category. If $\cal K$ is
abelian
and the sequence
$$ 0 \to K \to^k X \to^c C \to 0$$
is short exact, we show that
1) $K$ is finitely generated iff $c$ is finitely
presentable;
2) $k$ is finitely presentable iff $C$ is finitely
presentable. The ``if" directions fail for semi-abelian
varieties. We show that all but (possibly) 2)(if) follow from
analogous
properties which hold in all locally finitely presentable categories.
As for 2)(if), it holds as soon as $\cal K$ is also co-homological,
and
all its strong epimorphisms are regular. Finally, locally finitely
coherent
(resp. noetherian) abelian categories are characterized as those for which
all
finitely presentable morphisms have finitely generated (resp. presentable)
kernel objects.
Keywords: finitely presentable morphism, abelian category, Grothendieck category
2000 MSC: 18A20, 18E10, 18C35, 18E15
Theory and Applications of Categories,
Vol. 24, 2010,
No. 9, pp 209-220.
http://www.tac.mta.ca/tac/volumes/24/9/24-09.dvi
http://www.tac.mta.ca/tac/volumes/24/9/24-09.ps
http://www.tac.mta.ca/tac/volumes/24/9/24-09.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/9/24-09.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/9/24-09.ps