Homotopies of crossed complex morphisms of associative R-algebras

Abstract

Abstract In this study, given two crossed complexes 𝒞 {\mathcal{C}} and 𝒟 {\mathcal{D}} of associative R-algebras and a crossed complex morphism f : 𝒞 → 𝒟 {f\colon\mathcal{C}\to\mathcal{D}} , we construct a homotopy as a pair ( H , f ) {(H,f\/)} , where H = ( H n ) {H=(H_{n})} is a sequence of R-linear maps H n : C n → D n + 1 {H_{n}\colon C_{n}\to D_{n+1}} . Then we show that for a fixed pair 𝒞 {\mathcal{C}} and 𝒟 {\mathcal{D}} of crossed complexes of associative R-algebras, the family of all homotopies between crossed complex morphisms from 𝒞 {\mathcal{C}} to 𝒟 {\mathcal{D}} has a groupoid structure with crossed complex morphisms as objects and homotopies as morphisms.