G-kernels and Crossed Modules

We develop a unified framework based on topological crossed modules for various lifting obstructions for $\Gamma$-kernels. It allows us to identify actions, cocycle actions and $\Gamma$-kernels up to their natural equivalence relations with cohomology sets. The obstructions then appear as boundary maps in corresponding exact sequences. Since topological crossed modules are topological $2$-groups (in the categorical sense), they have classifying spaces, which come with a natural transformation from the cohomology to a homotopy set. For the crossed module that gives cocycle actions we prove a weak equivalence of the classifying space of the crossed module with one from bundle theory. In case the algebra is strongly self-absorbing we show that the homotopy set is a group and that the above natural transformation is a group isomorphism on an appropriate restriction of the cohomology set.