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

Contents

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).

Examples

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

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