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On the representability of actions in a semi-abelian category

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F. Borceux, G. Janelidze and G.M. Kelly

We consider a semi-abelian category V and we write Act(G,X) for
the set of actions of the object G on the object X, in the sense
of the theory of semi-direct products in V. We investigate the
representability of the functor Act(-,X) in the case where V
is locally presentable, with finite limits commuting with filtered
colimits. This contains all categories of models of a semi-abelian
theory in a Grothendieck topos, thus in particular all semi-abelian
varieties of universal algebra. For such categories, we
prove first that the representability of Act(-,X) reduces to
the preservation of binary coproducts. Next we give both a very
simple necessary condition and a very simple sufficient condition,
in terms of amalgamation properties, for the preservation of binary
coproducts by the functor Act(-,X) in a general semi-abelian category.
Finally, we exhibit the precise form of the more involved ``if and only
if'' amalgamation property corresponding to the representability of
actions: this condition is in particular related to a new notion of
``normalization of a morphism''. We provide also a wide supply of
algebraic examples and counter-examples, giving in particular
evidence of the relevance of the object representing Act(-,X),
when it turns out to exist.

Keywords:
semi-abelian category, variety, semi-direct product, action

2000 MSC:
18C10, 18D35, 18G15

*Theory and Applications of Categories,*
Vol. 14, 2005,
No. 11, pp 244-286.

http://www.tac.mta.ca/tac/volumes/14/11/14-11.dvi

http://www.tac.mta.ca/tac/volumes/14/11/14-11.ps

http://www.tac.mta.ca/tac/volumes/14/11/14-11.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/11/14-11.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/11/14-11.ps

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