#
nLab

model structure on symmetric spectra

Contents
### Context

#### Model category theory

**model category**

## Definitions

## Morphisms

## Universal constructions

## Refinements

## Producing new model structures

## Presentation of $(\infty,1)$-categories

## Model structures

### for $\infty$-groupoids

for ∞-groupoids

### for equivariant $\infty$-groupoids

### for rational $\infty$-groupoids

### for rational equivariant $\infty$-groupoids

### for $n$-groupoids

### for $\infty$-groups

### for $\infty$-algebras

#### general

#### specific

### 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

#### 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 $\mathcal{S}$-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

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.
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