nLab model structure on symmetric spectra

Idea

A model structure on spectra for symmetric spectra

The category of symmetric spectra is a presentation of the symmetric monoidal (∞,1)-category of spectra, with the special property that it implements the smash product of spectra such as to yield itself a symmetric monoidal model category of spectra: the model structure on symmetric spectra. This implies in particular that with respect to this symmetric smash product of spectra an E-∞ ring is presented simply as a plain commutative monoid in symmetric spectra.

Properties

Relation to model structures on sequential spectra

There is a Quillen equivalence to the Bousfield-Friedlander model structure on sequential spectra (HoveyShipleySmith 00, section 4.3, Mandell-May-Schwede-Shipley 01, theorem 0.1).

Relation to model structure on $\mathcal{S}$-modules

There is also a Quillen equivalence to the model structure on S-modules (Schwede 01)

Related concepts

model structure on spectra

• sequential spectrum, model structure on sequential spectra

with symmetric monoidal smash product of spectra

• symmetric spectrum, model structure on symmetric spectra

• orthogonal spectrum, model structure on orthogonal spectra

• S-module, model structure on S-modules

• excisive functor, model structure on excisive functors

• Model categories of diagram spectra

References

The projective and injective model structure on symmetric spectra are due to

• Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)

The “S-model structure” (also called “flat model structure” in Schwede 12, part III) is due to

• Brooke Shipley, A Convenient Model Category for Commutative Ring Spectra, 2003 (web)

See also

• Mark Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra 165 1 (2001) 63-127 [arXiv:0004051, doi:10.1016/S0022-4049(00)00172-9]

Another proof, beyond (Hovey-Shipley-Smith 00), of the Quillen equivalence to the Bousfield-Friedlander model structure is due to

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, theorem 0.1 of Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)

The Quillen equivalence to the model structure on S-modules is due to

• Stefan Schwede, S-modules and symmetric spectra, Math. Ann. 319, 517–532 (2001) (pdf)

Comprehensive review is in

• Stefan Schwede, part III of Symmetric spectra, 2012 (pdf)

Generalization to a model structure for parameterized spectra:

• Vincent Braunack-Mayer, Combinatorial parametrised spectra, Algebr. Geom. Topol. 21 (2021) 801-891 [arXiv:1907.08496, doi:10.2140/agt.2021.21.801]

  (based on the PhD thesis)