Adjunction between crossed modules of groups and algebras (Q2255536)

Crossed modules of groups are classical in homotopy theory. Generalizations, i.e., crossed modules of associative algebras appear in non-abelian cohomology and Hochschild cohomology. The authors study the relationship between the category of crossed modules of groups and the category of crossed modules of associative algebras. In particular, they construct an adjunction between the categories in which the left adjoint is analogous to the group algebra functor, and the right adjoint is analogous to the group of units functor. A consequence resulting easily from the general categorical relationships studied is Theorem 4.6: Theorem. The category of modules over a crossed module is isomorphic to the category of modules over its associated crossed algebra (along the adjunction).