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.