nLab
Bousfield localization of triangulated categories

Contents

Definition

Definition (triangulated subcategory)

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

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

Definition (Verdier quotient)

For ABA \subset B a triangulated subcategory, the Verdier quotient? category B/AB/A which is a triangulated category equipped with a canonical functor Q *:BB/AQ^*:B\to B/A that is also triangulated (additive and preserving distinguished triangles) and universal among all triangulated functors BDB\to D which send objects of AA to objects isomorphic to 00.

The Verdier quotient B/AB/A has the property that the only objects whose images in B/AB/A are isomorphic to the zero object are the objects from AA.

Definition (Bousfield localization)

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

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

B locB. 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 KK-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel (pdf)

Revised on October 17, 2012 20:46:49 by Zoran Škoda (161.53.130.104)