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

### Context

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

(∞,1)-category theory

## Models

#### Monoidal categories

monoidal categories

## In higher category theory

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The notion of commutative monoid (or commutative monoid object, commutative algebra, commutative algebra object) in a symmetric monoidal (infinity,1)-category is the (infinity,1)-categorical generalization of the notion of commutative monoid in a symmetric monoidal category. It is the commutative version of monoid in a monoidal (infinity,1)-category.

Note that commutative here really means $E_\infty$, in the sense of E-infinity operad.

## Definition

A commutative monoid in a symmetric monoidal (infinity,1)-category $C$ is a lax symmetric monoidal $(\infty,1)$-functor

$* \to C \,.$

In more detail, this means the following:

###### Definition

Given a symmetric monoidal (infinity,1)-category in its quasi-categorical incarnation as a coCartesian fibration of simplicial sets

$p : C^\otimes \to N(FinSet_*)$

a commutative monoid in $C$ is a section

$A : N(FinSet_*) \to C^\otimes$

such that $A$ carries collapsing morphisms in $FinSet_*$ to coCartesian morphisms in $C^\otimes$.

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

### Definition

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

### Properties

###### Theorem
• $CMon(C)$ 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 $CMon(C)$ and are computed in $C$ if $K$-colimits exist in $C$ are preserved by tensor product with any object.

• $CMon(C)$ has all (∞,1)-limits and these are computed in $C$.

This is (Lurie DAG III, section 4) or (Lurie HA, sections 3.2.2 and 3.2.3).

###### Corollary

$(\infty,1)$-Colimits over simplicial diagrams exists in $CMon(C)$ 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

An equivalent reformulation of commutative monoids in terms (∞,1)-algebraic theories is in

• James Cranch, Algebraic Theories and $(\infty,1)$-Categories (arXiv)

Revised on March 9, 2015 12:35:40 by Adeel Khan (93.131.147.91)