“…The objects in Crs n are homotopy n-types, that is, they have trivial "homotopy groups" in dimensions greater than n. The homotopy groups of a crossed complex are defined as follows: π 0 (C) is the set of connected components of the base groupoid, so π 0 (C) = π 0 (G). Similarly, π 1 (C) = π 1 (C 2 ) = G/ im(δ 2 ), the fundamental groupoid of the base crossed module of C. For n 2, π n (C) is defined as the "homology group" H n (C) : π 1 (C) → Ab of the induced chain complex of π 1 (C)-modules (9). Note that if we consider π n (C) as a G-module via the canonical projection q : G → π 1 (C), for n 2, the G-crossed module (π n (C), 0) is the kernel of the induced map ∂ n : C n / im(∂ n+1 ) → C n−1 (see (10) below).…”