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On categorical crossed modules

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P. Carrasco, A.R. Garzon and E.M. Vitale

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.

http://www.tac.mta.ca/tac/volumes/16/22/16-22.dvi

http://www.tac.mta.ca/tac/volumes/16/22/16-22.ps

http://www.tac.mta.ca/tac/volumes/16/22/16-22.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/22/16-22.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/22/16-22.ps

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