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:

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

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 $S\in S_0$.

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 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.
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