A triangulated subcategory is called thick if with any object in it contains all its direct summands in .
For a triangulated subcategory, the Verdier quotient? category which is a triangulated category equipped with a canonical functor that is also triangulated (additive and preserving distinguished triangles) and universal among all triangulated functors which send objects of to objects isomorphic to .
The Verdier quotient has the property that the only objects whose images in are isomorphic to the zero object are the objects from .
Given a thick subcategory , we say that the Bousfield localization exists if the Verdier quotient functor has a right adjoint functor which is then (automatically) triangulated and fully faithful.
To amplify, writing a Bousfield localization of a triangulated category is in particular an adjunction
Compare this to Bousfield localization of model categories, noticing that most triangulated categories arise as homotopy categories of stable (∞,1)-categories, hence of homotopy categories of the model categories presenting these.
Also note, to a tensored, triangulated category, one may associate a Bousfield lattice which has deep connections to the above topic.
Amnon Neeman, Triangulated categories, chapter 9
Amnon Neeman, The connection between the -theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel (pdf)