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

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

• A commutative monoid in ∞Grpd is an E-∞ space.

• A commutative monoid in the stable (infinity,1)-category of spectra is a commutative ring spectrum or E-infinity ring.

Related concepts

• monoid in a monoidal (infinity,1)-category

• infinity-algebra over an (infinity,1)-operad

• commutative monoid in a symmetric monoidal category

• module over a monoid

• ∞-group completion

References

• Jacob Lurie, Commutative Algebra.

• Jacob Lurie, Higher Algebra.

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

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