Research Information System
Applied Categorical Structures, vol.29, no.5, pp.827-847, 2021 (SCI-Expanded)
In this work, given two crossed modules M= (μ: M → A) and N= (η: N → B) of R-algebroids and a crossed module morphism f: M→ N, we introduce an f-derivation as an ordered pair H= (H1, H) of maps H1: Mor (A) → Mor (N) and H: A → Mor (B) which are subject to satisfy certain axioms and show that f and H determine a crossed module morphism g: M→ N. Then calling such a pair (H, f) a homotopy from f to g we prove that there exists a groupoid structure of which objects are crossed module morphisms from M to N and morphisms are homotopies between crossed module morphisms. Moreover, given two crossed module morphisms f, g: M→ N, we introduce an fg-map as a map Λ: A → Mor (N) subject to some conditions and then show that Λ determines for each homotopy (H, f) from f to g a homotopy (H′, f) from f to g. Furthermore, calling such a pair (Λ, (H, f)) a 2-fold homotopy from (H, f) to (H′, f) we prove that the groupoid structure constructed by crossed module morphisms from M to N and homotopies between them is upgraded by 2-fold homotopies to a 2-groupoid structure. Besides, in order to see reduced versions of all general constructions mentioned, we examine homotopies of crossed modules of associative R-algebras, as a pre-stage.