Localisation and colocalisation of triangulated categories at thick subcategories

Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a natural long exact sequence containing the canonical map between a homological functor and its total derived functor with respect to a thick subcategory.

💡 Research Summary

The paper investigates a symmetric counterpart to the classical localisation of a triangulated category at a thick subcategory. Let 𝒯 be a triangulated category and 𝒮⊂𝒯 a thick subcategory (i.e. closed under direct summands, shifts, and extensions). The usual localisation 𝒯/𝒮 is obtained by formally inverting all morphisms whose cone lies in 𝒮; this kills the 𝒮‑information and retains only the 𝒮‑acyclic part of the homological data. The author introduces a new construction, called colocalisation and denoted 𝒯\𝒮, which instead “quotients out” the 𝒮‑acyclic part while preserving the 𝒮‑information.

The definition proceeds by considering 𝒮‑exact triangles: a distinguished triangle X→Y→Z→X