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.
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 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 $\vert J\vert \lt \alpha$.