# nLab model structure on spectra

model category

## Model structures

for ∞-groupoids

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

#### Stable homotopy theory

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

• Aldridge Bousfield, Eric Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

(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

• Lydakis, section 10 of Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)

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

• William Dwyer, Localizations, In Axiomatic, enriched and motivic homotopy theory, volume 131 of NATO Sci. Ser. II Math. Phys. Chem., pages 3–28. Kluwer Acad. Publ., Dordrecht, 2004