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 $XAss \simeq Cat^1-Ass$ 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.
Keywords: split extension classifier, category of interest, associative algebra, crossed module, cat^1-associative algebra, equivalence of categories, actor, universal strict general actor, bimultiplier
2010 MSC: 08A99, 08C05, 16E99, 18B99
Theory and Applications of Categories, Vol. 30, 2015, No. 25, pp 882-908.
Revised 2016-06-20. Original version at