nLab
model structure on symmetric spectra

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

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

general

specific

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

Contents

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 𝒮\mathcal{S}-modules

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

model structure on spectra

with symmetric monoidal smash product of spectra

References

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

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

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

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

Last revised on June 16, 2016 at 05:11:56. See the history of this page for a list of all contributions to it.