#
nLab

model structure on S-modules

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 $n$-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 S-modules.

The category of *S-modules* (EKMM 97) 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 S-modules.

## Properties

### Relation to the model structure on symmetric spectra

There is a Quillen equivalence to the model structure on symmetric spectra (Schwede 01).

### Relation to model structure on orthogonal structure

Comparison to the model structure on orthogonal spectra is due to (Mandell 04).

model structure on spectra

with symmetric monoidal smash product of spectra

## References

The construction originates in

Review includes

Comparison to the model structure on symmetric spectra is due to

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

Comparison to the model structure on orthogonal spectra is due to

Last revised on March 10, 2019 at 14:40:14.
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