nLab
stable model category

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Stable homotopy theory

stable homotopy theory

Ingredients

Contents

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 C is

Properties

Characterization

Proposition

Let C be a stable model category that is in addition

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

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

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

This is theorem 3.3.3 in (Schwede-Shipley)

Remark

Notice the similarity (but superficial difference: sSet/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.

References

Revised on August 28, 2012 01:43:27 by Urs Schreiber (89.204.130.6)
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