# nLab model structure on symmetric spectra

Contents

### Context

#### Model category theory

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 $\infty$-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

#### Stable homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

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 structure on $\mathcal{S}$-modules

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

model structure on 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)