# nLab commutative monoid in a symmetric monoidal (infinity,1)-category

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

## Models

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

## In higher category theory

#### Higher algebra

higher algebra

universal algebra

# Contents

## $\left(\infty ,1\right)$-Category of commutative monoids

### Definition

For $C$ a symmetric monoidal (∞,1)-category write $\mathrm{CMon}\left(C\right)$ for the $\left(\infty ,1\right)$-category of commutative monoids in $C$.

### Properties

###### Theorem
• $\mathrm{CMon}\left(C\right)$ has all (∞,1)-coproducts and these are computed as tensor products in $C$.

• For $K$ a sifted (infinity,1)-category , (∞,1)-colimits of shape $K$ exist in $\mathrm{CMon}\left(C\right)$ and are computed in $C$ if $K$-colimits exist in $C$ are preserved by tensor product with any object.

• $\mathrm{CMon}\left(C\right)$ has all (∞,1)-limits and these are computed in $C$.

This is (Lurie, section 4).

###### Corollary

$\left(\infty ,1\right)$-Colimits over simplicial diagrams exists in $\mathrm{CMon}\left(C\right)$ and are computed in $C$ if they exist in $C$ and a preserved by tensor products.

Because the simplex category is a sifted (infinity,1)-category (as discussed there).

## References

Created on December 2, 2010 20:05:49 by Urs Schreiber (131.211.232.152)