nLab
well-generated triangulated category

Contents

Idea

A well-generated triangulated category is a strengthening of the notion of compactly generated triangulated category which was introduced by Neeman, 2001. The following definition is from (Krause) and is somewhat shorter and more natural than Neeman’s original definition.

Definition

Definition

Let TT be a triangulated category with arbitrary coproducts. Then TT is well-generated in the sense of Neeman if and only if there exists a set S 0S_0 of objects satisfying:

  1. an object XX of TT is zero if [S,X]=0[S,X]=0 for all SS 0S\in S_0;

  2. for every set of maps X iY iX_i\to Y_i in TT, the induced map [S, IX i][S, IY i][S,\coprod_I X_i]\to[S,\coprod_I Y_i] is surjective for all SS 0S\in S_0 whenever [S,X i][S,Y i][S,X_i]\to[S,Y_i] is surjective for all ii and all SS 0S\in S_0.

  3. the objects of S 0S_0 are α\alpha-small for some cardinal α\alpha.

We recall that to say an object SS is α\alpha-small in a triangulated category is to say that every map S JX jS\to\coprod_J X_j factors through some S JX jS\to\coprod_J X_j whenever J<α\vert J\vert \lt \alpha.

References

  • Henning Krause, On Neeman’s Well Generated Triangulated Categories, Documenta Mathenatica 6 (2001) (pdf).
  • Amnon Neeman, Triangulated Categories, Annals of Mathematics Studies 148, Princeton University Press (2001).

Revised on September 23, 2013 00:05:36 by Anonymous Coward (108.240.248.132)