nLab
stable model category

Idea

A stable model category is a 1-category structure used to present a stable (∞,1)-category in analogy to how a general model category encodes a general (∞,1)-category.

Defintion

A stable model category $C$ is

• a pointed

• model category

• such that the loop space object functor $\Omega$ and the suspension functor $\Sigma$, are inverse equivalences on the homotopy category $\mathrm{Ho}(C)$:

  $$\Omega :\mathrm{Ho}(C)\stackrel{{\leftarrow }^{\simeq }}{{\to }^{\simeq }}:\mathrm{Ho}(C):\Sigma$$\Omega : Ho(C) \stackrel{\leftarrow^\simeq}{\to^\simeq} : Ho(C) : \Sigma

Properties

Proposition

Let $C$ be a stable model category that is in addition

• a simplicial model category;

• a cofibrantly generated model category;

• a proper model category;

• with a set $S$ of compact generators;

then there is a chain of SSet Quillen equivalences linking $C$ to the the spectrum-enriched functor category

$$C\simeq \mathrm{Sp}\mathrm{Cat}(ℰ(S),\mathrm{Sp})$$C \simeq Sp Cat(\mathcal{E}(S), Sp)

equipped with the global model structure on functors, where $ℰ(S)$ is the $\mathrm{Sp}$-enriched category given by…

This is theorem 3.3.3 in ClassStabMod.

Remark Notice the similarity (but superficial difference: $\mathrm{SSet}$/$\mathrm{Sp}$-enrichment localization/no-localization) to the stable Giraud theorem discussed at stable (∞,1)-category.

References

• Stefan Schwede, Brooke Shipley, Classification of stable model categories (pdf)