ACTIONS IN MODIFIED CATEGORIES OF INTEREST WITH APPLICATION TO CROSSED MODULES

The existence of the split extension classifier of a crossed module in the category of associative algebras is investigated. According to the equivalence of categories XAss similar or equal to Cat(1)-Ass we consider this problem in Cat(1)-Ass. This category is not a category of interest, it satisfies its all axioms except one. The action theory developed in the category of interest is adapted to the new type of category, which will be called modified category of interest. Applying the results obtained in this direction and the equivalence of categories we find a condition under which there exists the split extension classifier of a crossed module and give the corresponding construction.