# nLab compactly generated triangulated category

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

### Theorems

#### Stable homotopy theory

stable homotopy theory

# Contents

#### Compact objects

objects $d\in C$ such that $C\left(d,-\right)$ commutes with certain colimits

# Contents

## 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 $𝒯\left({\Sigma }^{m}G,X\right)=0$ for all $G\in 𝒢$ and $m\in ℤ$, then $X=0$.

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

$\coprod _{i\in I}𝒯\left(G,{X}_{i}\right)\to 𝒯\left(G,\coprod _{i\in I}{X}_{i}\right)$\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.

## References

Revised on August 29, 2012 12:17:15 by Karol Szumiło? (131.220.132.179)