nLab
well-generated triangulated category

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 $T$ 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 to say an object $S$ is $\alpha$-small in a triangulated category is to say that every map $S\to {\coprod }_J{X}_j$ factors through some $S\to {\coprod }_J{X}_j$ whenever $\mid J\mid <\alpha$.

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).