nLab Bousfield localization of triangulated categories

Definition

Definition (triangulated subcategory)

A triangulated subcategory $A$ of a triangulated category $B$ is a nonempty subcategory closed under suspension of objects and such that for all objects $X,Y$ in $A$ if $X\to Y\to Z\to X[1]$ is a distinguished triangle in $B$, then $Z$ is in $A$.

A triangulated subcategory $A$ is called thick if with any object in $A$ it contains all its direct summands in $B$.

Definition (Verdier quotient)

For $A \subset B$ a triangulated subcategory, the Verdier quotient? category $B/A$ which is a triangulated category equipped with a canonical functor $Q^*:B\to B/A$ that is also triangulated (additive and preserving distinguished triangles) and universal among all triangulated functors $B\to D$ which send objects of $A$ to objects isomorphic to $0$.

The Verdier quotient $B/A$ has the property that the only objects whose images in $B/A$ are isomorphic to the zero object are the objects from the thickening of $A$ (the smallest thick subcategory containing $A$).

Definition (Bousfield localization)

Given a thick subcategory $A\subset B$, we say that the Bousfield localization exists if the Verdier quotient functor $Q^*$ has a right adjoint functor $Q_*$ which is then (automatically) triangulated and fully faithful.

To amplify, writing $B_{loc} := B/A$ a Bousfield localization of a triangulated category $B$ is in particular an adjunction

$$B_{loc} \stackrel{\stackrel{}{\leftarrow}}{\hookrightarrow} B \,.$$

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.

References

• Henning Krause, Localization theory for triangulated categories, arXiv/0806.1324

• Amnon Neeman, Triangulated categories, chapter 9

• Amnon Neeman, The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel (pdf)