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Internal crossed modules and Peiffer Condition

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Sandra Mantovani, Giuseppe Metere

In this paper we show that in a homological category in the sense
of F. Borceux and D. Bourn, the notion of an internal precrossed
module corresponding to a star-multiplicative graph, in the sense
of G. Janelidze, can be obtained by directly internalizing the
usual axioms of a crossed module, via equivariance. We then
exhibit some sufficient conditions on a homological category under
which this notion coincides with the notion of an internal crossed
module due to G. Janelidze. We show that this is the case for any
category of distributive $\Omega_2$-groups, in particular for the
categories of groups with operations in the sense of G. Orzech.

Keywords:
internal crossed module, reflexive graph, internal
action, semiabelian category

2000 MSC:
18D35, 18G50, 20L05, 20J15

*Theory and Applications of Categories,*
Vol. 23, 2010,
No. 6, pp 113-135.

http://www.tac.mta.ca/tac/volumes/23/6/23-06.dvi

http://www.tac.mta.ca/tac/volumes/23/6/23-06.ps

http://www.tac.mta.ca/tac/volumes/23/6/23-06.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/6/23-06.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/6/23-06.ps

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