Geodesic
On categorical crossed modules
Theory and applications of categories, Tome 16 (2006), pp. 585-618.

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The well-known notion of crossed module of groups is raised in this paper to the categorical level supported by the theory of categorical groups. We construct the cokernel of a categorical crossed module and we establish the universal property of this categorical group. We also prove a suitable 2-dimensional version of the kernel-cokernel lemma for a diagram of categorical crossed modules. We then study derivations with coefficients in categorical crossed modules and show the existence of a categorical crossed module given by inner derivations. This allows us to define the low-dimensional cohomology categorical groups and, finally, these invariants are connected by a six-term 2-exact sequence obtained by using the kernel-cokernel lemma.
Classification : 18D10, 18G50, 20J05, 20L05
Keywords: crossed module, categorical group, categorical crossed module, derivation, 2-exact sequence, cohomology categorical group 
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     volume = {16},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2006_16_a21/}
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P. Carrasco; A.R. Garzon; E.M. Vitale. On categorical crossed modules. Theory and applications of categories, Tome 16 (2006), pp. 585-618. http://geodesic.mathdoc.fr/item/TAC_2006_16_a21/