nLab
model structure on spectra

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

• model structure on sequential spectra

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

• Bousfield-Friedlander model structure,

with its analogue in topological spaces, the

• model structure on topological sequential spectra.

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

• model structure on combinatorial spectra.

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

• model structure for excisive functors.

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

• model structure on orthogonal spectra,

• model structure on symmetric spectra,

A unified treatment and comparison of these is in

• Model categories of diagram spectra

Then there is also the

• model structure on S-modules.

Related concepts

• model structure for connective spectra

• model structure on presheaves of spectra

• model structure for ring spectra

• model structure for dendroidal left fibrations

• higher algebra, brave new algebra, higher linear algebra

• stable (infinity,1)-category

• Spectra

References

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

• 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

See also

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)

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

• Mark Hovey, Spectra and symmetric spectra in general model categories (arXiv:0004051)

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

• Bjørn Ian Dundas, Oliver Röndigs, Paul Arne Østvær, Enriched functors and stable homotopy theory, Doc. Math., 8:409–488, 2003 (EuDML)

Review:

• Daniel Dugger, Stable categories and spectra via model categories (pdf), in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds.), Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press