Homotopy categories and idempotent completeness, weight structures and weight complex functors

Abstract: This article provides some basic results on weight structures, weight complex functors and homotopy categories. We prove that the full subcategories K(A)^{w < n}, K(A)^{w > n}, K(A)^- and K(A)^+ (of objects isomorphic to suitably bounded complexes) of the homotopy category K(A) of an additive category A are idempotent complete, which confirms that (K(A)^{w <= 0}, K(A)^{w >= 0}) is a weight structure on K(A). We discuss weight complex functors and provide full details of an argument sketched by M. Bondarko, which shows that if w is a bounded weight structure on a triangulated category T that has a filtered triangulated enhancement T' then there exists a strong weight complex functor T -> K(heart(w))^{anti}. Surprisingly, in order to carry out the proof, we need to impose an additional axiom on the filtered triangulated category T' which seems to be new.