A triangulated subcategory of a triangulated category is a nonempty subcategory closed under suspension of objects and such that for all objects in if is a distinguished triangle in , then is in .
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 the thickening of (the smallest thick subcategory containing ).
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.
Henning Krause, Localization theory for triangulated categories, arXiv/0806.1324
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)
Last revised on February 27, 2024 at 16:58:17. See the history of this page for a list of all contributions to it.