nLab
compactly generated triangulated category

Definition

Definition

Let $𝒯$ be a triangulated category with coproducts. Then $𝒯$ is compactly generated if there is a set $𝒢$ of objects of $𝒯$ such that

1. Whenever $X$ is an object such that $𝒯({\Sigma }^{m}G,X)=0$ for all $G\in 𝒢$ and $m\in \mathbb{Z}$, then $X=0$.

2. All objects in $𝒢$ are compact i.e. for all $G\in 𝒢$ and for every family $\{X_i\mid i\in I\}$ of objects of $𝒯$

   $$\coprod _{i\in I}𝒯(G,X_i)\to 𝒯(G,\coprod _{i\in I}{X}_i)$$\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 $𝒯$ is a triangulated category with coproducts, then a set of objects $𝒢$ satisfies the two conditions of def. 1 if and only if the smallest localizing subcategory of $𝒯$ that contains $𝒢$ is $𝒯$ itself.

This is Lemma 2.2.1. of (Schwede-Shipley).

Brown representability theorem holds in compactly generated triangulated categories.

Theorem

If $𝒯$ is a compactly generated triangulated category and $H:{𝒯}^{\mathrm{op}}\to \mathrm{Ab}$ 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

References

• Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules. Topology 42 (2003), no. 1, 103–153.