nLab
model structure on spectra

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-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

Contents

Idea

A model category structure on a category of spectra presents a stable (∞,1)-category of spectrum objects.

Typically and naturally, a model structure on spectra forms a stable model category. In good cases it also forms a symmetric monoidal model category with respect to the smash product of spectra, see at symmetric monoidal smash product of spectra.

Examples

A classical

in simplicial sets, not however with a symmetric monoidal smash product, is (Bousfield-Friedlander 78) the

with its analogue in topological spaces, the

These are related by a zig-zag of Quillen equivalences to the

A Quillen equivalent model structure to the model structures on sequential spectra that does carry a symmetric monoidal smash product of spectra is the

This models spectra as enriched functors on the site of pointed finite homotopy types. Restricting that to smaller sub-sites, yields model structures for “highly structured spectra” with a symmetric monoidal smash product of spectra: the

A unified treatment and comparison of these is in

Then there is also the

References

The Bousfield-Friedlander model structure on sequential spectra in simplicial sets is due to

(a quick review of this is in Lydakis 98, section 10).

The Quillen equivalent model structure on excisive functors on pointed simplicial sets is due to

and a similar model structure for functors on topological spaces has been given in

See also

A discussion of model structures on spectra in general ambient model categories (general spectrum objects, including e.g. motivic spectra) is in

and for the Bousfield-Friedlander-type model structure in

and for the excisive-functor-type model structure in

Review: