Homotopies of crossed complex morphisms of associative R-algebras

Abstract

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

Homotopies of Crossed Modules of R-Algebroids

“…meaning that (H (K L)) 1 = ((H K ) L)1 . So, H (K L) = (H K ) L and consequently(H , f ) ((K , g) (L, h)) = (H , f ) (K L, g) = (H (K L) , f ) = ((H K ) L, f ) = (H K , f ) (L, h) = ((H , f ) (K , g)) (L, h) ,which means that the composition is associative.…”