Topological Crossed Semimodules and Schreier Internal Categories in the Category of Topological Monoids

The aim of this paper is to introduce the notion of Schreier internal categories in the category of topological monoids and of topological crossed semimodules and to prove the categorical equivalence between them. This is the generalization of equivalence between the category of internal categories in the category of topological groups and the category of topological crossed modules. Moreover, we obtained a Schreier internal category as a special sort of 2-category with one object in the category of topological monoids.