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 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.
Internal crossed modules and Peiffer condition / S. Mantovani, G. Metere. - In: THEORY AND APPLICATIONS OF CATEGORIES. - ISSN 1201-561X. - 23:6(2010), pp. 113-135.
Internal crossed modules and Peiffer condition
S. MantovaniPrimo
;G. MetereUltimo
2010
Abstract
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 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.Pubblicazioni consigliate
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