nLab compactly generated triangulated category

Definition

Definition

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

1. 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 $X = 0$.

2. 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 $\mathcal{T}$

   $$\coprod_{i \in I} \mathcal{T}(G, X_i) \to \mathcal{T}\Big(G,\coprod_{i \in I} X_i\Big)$$

   is an isomorphism.

Properties

Proposition

If $\mathcal{T}$ is a triangulated category with coproducts, then a set of objects $\mathcal{G}$ satisfies the two conditions of def. 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.

Theorem

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.

Examples

• 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 $R$.

Related concepts

• well-generated triangulated category

• compactly generated model category

• compactly generated (∞,1)-category

References

• 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.