Verdier quotients of homotopy categories

TL;DR

This paper investigates Verdier quotients of various homotopy categories derived from an additive subcategory of an abelian category, analyzing their relationships through localization sequences and equivalences.

Contribution

It provides a systematic study of Verdier quotients of homotopy categories, establishing conditions for equivalences and describing their structural relationships.

Findings

Many Verdier quotients are equivalent categories.

Localization sequences and recollement diagrams are constructed.

The study covers various boundedness conditions of complexes.

Abstract

We study Verdier quotients of diverse homotopy categories of a full additive subcategory $\mathcal E$ of an abelian category. In particular, we consider the categories $K^{x,y}({\mathcal E})$ for $x\in\{\infty, +,-,b\}$, and $y\in\{\emptyset,b,+,-,\infty\}$ the homotopy categories of left, right, unbounded complexes with homology being $0$, bounded, left or right bounded, or unbounded. Inclusion of these categories give a partially ordered set, and we study localisation sequences or recollement diagrams between the Verdier quotients, and prove that many quotients lead to equivalent categories.