(also 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.
Last revised on February 10, 2014 at 06:54:30. See the history of this page for a list of all contributions to it.