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.
Keywords: crossed module, categorical group, categorical crossed module, derivation, 2-exact sequence, cohomology categorical group
2000 MSC: 18D10, 18G50, 20J05, 20L05
Theory and Applications of Categories,
Vol. 16, 2006,
No. 22, pp 585-618.