model structure on spectra


Model category theory

model category



Universal constructions


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



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



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.


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


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

  • Aldridge Bousfield, Eric Friedlander, Homotopy theory of Γ\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

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

  • Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 77-104 (pdf)

and for the excisive-functor-type model structure in

Revised on April 25, 2016 06:57:48 by Urs Schreiber (