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

TL;DR

This paper establishes foundational results on weight structures, homotopy categories, and weight complex functors, proving idempotent completeness and constructing strong weight complex functors under new axioms.

Contribution

It proves idempotent completeness of certain subcategories of homotopy categories and constructs strong weight complex functors with a new additional axiom.

Findings

Idempotent completeness of subcategories of homotopy categories

Existence of strong weight complex functors under new axioms

Confirmation that (K(A)^{w <= 0}, K(A)^{w >= 0}) forms a weight structure

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.