# nLab stable model category

model category

## Model structures

for ∞-groupoids

### for $\left(\infty ,1\right)$-sheaves / $\infty$-stacks

#### Stable homotopy theory

stable homotopy theory

# Contents

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## 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

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

$\Omega :\mathrm{Ho}\left(C\right)\stackrel{\stackrel{sime}{←}}{\underset{\simeq }{\to }}:\mathrm{Ho}\left(C\right):\Sigma \phantom{\rule{thinmathspace}{0ex}}.$\Omega : Ho(C) \stackrel{\overset{\sime}{\leftarrow}}{\underset{\simeq}{\to}} : Ho(C) : \Sigma \,.

## Properties

### Characterization

###### Proposition

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

• with a set $S$ of compact generators;

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

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

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

This is theorem 3.3.3 in (Schwede-Shipley)

###### 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.

Moreover, by Schwede-Shipley 03 theorems, 3.1.1, 3.3.3, 3.8.2 stable model categories equivalent (by zig-zags of Quillen equivalences) to categories of module spectra over some ring spectrum. If that is an Eilenberg-MacLane spectrum, then this identifies the corresponding stable model categories with the model structure on unbounded chain complexes.

### Relation to stable $\infty$-categories

Stabilization of model categories is a model for the abstractly defined stabilization in (infinity,1)-category theory (Robalo 12, prop. 4.15).

## References

The standard accounts are

Discussion of the notion of stable model categories with the abstract notion of stabilization in (infinity,1)-category theory is in section 4.2 (prop. 4.15) of

• Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, June 2012 (arxiv:1206.3645)

Revised on October 2, 2013 11:43:06 by Urs Schreiber (82.169.114.243)