The ternary commutator obstruction for internal crossed modules

Details The ternary commutator obstruction for internal crossed modules The ternary commutator obstruction for internal crossed modules

2011-07-05

arxiv.org/abs/1107.0954v1

Mathematics

Category Theory

48 pages

Scientific paper

We study the notion of internal crossed module in terms of cross-effects of the identity functor. These cross-effects give rise to a concept of commutator which allows a description of internal categories, (pre)crossed modules, Beck modules, and abelian extensions in finitely cocomplete homological categories in a way which is very close to the case of groups. We single out the obstruction which prevents a Peiffer graph being a groupoid - which in a semi-abelian context is known to vanish precisely when the Smith is Huq condition holds, so is invisible in the category of groups - as a certain ternary commutator. Such a ternary commutator appears in the Hopf formula for the third homology with coefficients in the abelianisation functor and in the interpretation of the second cohomology of an object with coefficients in a module. It is generally not decomposable into nested binary commutators: this happens, for instance, in the category of loops, where Smith is Huq is shown not to hold.

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