model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
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.
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).
There is also a Quillen equivalence to the model structure on S-modules (Schwede 01)
with symmetric monoidal smash product of spectra
symmetric spectrum, model structure on symmetric spectra
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
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
Comprehensive review is in
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)
Last revised on April 1, 2023 at 13:36:15. See the history of this page for a list of all contributions to it.