From Simplicial Homotopy to Crossed Module Homotopy in Modified Categories of Interest

Abstract: We address the (pointed) homotopy of crossed module morphisms in modified categories of interest; which generalizes the groups and various algebraic structures. We prove that, the homotopy relation gives rise to an equivalence relation; furthermore a groupoid structure, without any restriction on neither domain nor co-domain of the crossed module morphism. Additionally, we consider the particular cases such as associative algebras, Leibniz algebras, Lie algebras and dialgebras of crossed modules of this generalized homotopy definition. Then as the main part of the paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy and also the homotopy equivalence.