nLab
compactly generated triangulated category

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Stable homotopy theory

Compact objects

Contents

Definition

Definition

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

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

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

    iI𝒯(G,X i)𝒯(G, iIX 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\mathcal{T} is a triangulated category with coproducts, then a set of objects 𝒢\mathcal{G} satisfies the two conditions of def. 1 if and only if the smallest localizing subcategory of 𝒯\mathcal{T} that contains 𝒢\mathcal{G} is 𝒯\mathcal{T} itself.

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

Brown representability theorem holds in compactly generated triangulated categories.

Theorem

If 𝒯\mathcal{T} is a compactly generated triangulated category and H:𝒯 opAbH : \mathcal{T}^\mathrm{op} \to Ab is a cohomological functor, then HH is representable.

Examples

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.

Revised on February 10, 2014 06:54:30 by Urs Schreiber (89.204.135.153)