Homotopies of crossed complex morphisms of associative \(R\)-algebras
Group crossed modules were originally introduced by \textit{J. H. C. Whitehead} [Ann. Math. (2) 42, 409--428 (1941; Zbl 0027.26404)] as algebraic models for homotopy \(2\)-types. The notion of crossed complex goes back to \textit{A. L. Blakers} [Ann. Math. (2) 49, 428--461 (1948; Zbl 0040.25701)] and \textit{J. H. C. Whitehead} [Bull. Am. Math. Soc. 55, 453--496 (1949; Zbl 0040.38801)] under different names. \par The notion of crossed modules of algebras was generalized to crossed complexes over algebras by \textit{T. Porter} [J. Algebra 99, 458--465 (1986; Zbl 0594.13011)]. The authors study the homotopy theory of morphisms between crossed complexes of associative \(R\)-algebras for a commutative ring \(R\). The crossed module case result by \textit{I. I. Akça} et al. [Homology Homotopy Appl. 18, No. 1, 99--128 (2016; Zbl 1355.18005)] is extended and it is proved that if \(\mathcal{C}\) and \(\mathcal{D}\) are two crossed complexes of associative \(R\)-algebras, then there is a groupoid of crossed complex morphisms \(\mathcal{C}\to \mathcal{D}\) and their homotopies. Furthermore, it is shown that the homotopy relations between crossed complex morphisms is an equivalence relation.
scientific article; zbMATH DE number 7339601
scientific article