nLab
well-generated triangulated category

Contents

Idea

A well-generated triangulated category is a generalization 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 an object SS in a triangulated category is α\alpha-small if every map S JX jS\to\coprod_J X_j factors through IX j\coprod_I X_j for some IJI \subseteq J with |I|<α\vert I\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).

Last revised on June 20, 2018 at 14:58:43. See the history of this page for a list of all contributions to it.