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E-infinity monoid in a symmetric monoidal model category

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

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Higher algebra

Contents

Idea

An E E_\infty-monoid (or E-infinity monoid object) in a symmetric monoidal model category CC is an algebra over an operad over the E-infinity operad. Assuming that CC is cofibrantly generated, there is a model structure on E E_\infty-monoids, given by the model structure on algebras over an operad over the E-infinity operad (which is cofibrant). This model category presents the (infinity,1)-category of commutative monoids in a symmetric monoidal (infinity,1)-category (in the symmetric monoidal (infinity,1)-category presented by CC).

Rectification

In some symmetric monoidal model categories, E E_\infty-monoids can be rectified to (strictly) commutative monoids in a symmetric monoidal model category. See there for more.

References

Created on March 11, 2015 at 13:15:32. See the history of this page for a list of all contributions to it.