nLab well-generated triangulated category

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

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

2. for every set of maps $X_i\to Y_i$ in $T$, the induced map $[S,\coprod_I X_i]\to[S,\coprod_I Y_i]$ is surjective for all $S\in S_0$ whenever $[S,X_i]\to[S,Y_i]$ is surjective for all $i$ and all $S\in S_0$.

3. the objects of $S_0$ are $\alpha$-small for some cardinal $\alpha$.

We recall that an object $S$ in a triangulated category is $\alpha$-small if every map $S\to\coprod_J X_j$ factors through $\coprod_I X_j$ for some $I \subseteq J$ with $\vert I\vert \lt \alpha$.

References

• Henning Krause, On Neeman’s well generated triangulated categories, Documenta Mathematica 6 (2001) (pdf).

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

• Amnon Neeman, Triangulated Categories, Annals of Mathematics Studies 148, Princeton University Press (2001).