model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
(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: