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

TL;DR

This paper develops a generalized homotopy theory for crossed modules within modified categories of interest, establishing an equivalence relation and a groupoid structure, and demonstrates its preservation under a functor from simplicial objects.

Contribution

It introduces a new homotopy framework for crossed modules in modified categories of interest, extending classical concepts to broader algebraic structures.

Findings

Homotopy relation forms an equivalence relation.

A groupoid structure is established for crossed module homotopies.

Homotopy preservation under the functor from simplicial objects to crossed modules.

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.