and
nonabelian homological algebra
Let $\mathcal{T}$ be a triangulated category with coproducts. Then $\mathcal{T}$ is compactly generated if there is a set $\mathcal{G}$ of objects of $\mathcal{T}$ such that
Whenever $X$ is an object such that $\mathcal{T}(\Sigma^m G, X) = 0$ for all $G \in \mathcal{G}$ and $m \in \mathbb{Z}$, then $X = 0$.
All objects in $\mathcal{G}$ are compact i.e. for all $G \in \mathcal{G}$ and for every family $\{ X_i \mid i \in I\}$ of objects of $\mathcal{T}$
is an isomorphism.
If $\mathcal{T}$ is a triangulated category with coproducts, then a set of objects $\mathcal{G}$ satisfies the two conditions of def. 1 if and only if the smallest localizing subcategory of $\mathcal{T}$ that contains $\mathcal{G}$ is $\mathcal{T}$ itself.
This is Lemma 2.2.1. of (Schwede-Shipley).
Brown representability theorem holds in compactly generated triangulated categories.
If $\mathcal{T}$ is a compactly generated triangulated category and $H : \mathcal{T}^\mathrm{op} \to Ab$ is a cohomological functor, then $H$ is representable.
The sphere spectrum is a compact generator for the stable homotopy category.
More generally, if $R$ is a ring spectrum, then the homotopy category of module spectra over $R$ is compactly generated by $R$.
Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules. Topology 42 (2003), no. 1, 103–153.
Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, 148. Princeton University Press, 2001.