nLab
stable model category

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Stable homotopy theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

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 CC is

Properties

Characterization

Proposition

Let CC be a stable model category that is in addition

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

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

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

This is theorem 3.3.3 in (Schwede-Shipley)

Remark

Notice the similarity (but superficial difference: sSetsSet/SpSp-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 March 17, 2014 01:43:47 by Anonymous Coward (138.37.82.192)