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

Context

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

Monoidal categories

Higher 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 E_\infty, in the sense of E-infinity operad.

Definition

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

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

a commutative monoid in CC is a section

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

such that AA carries collapsing morphisms in FinSet *FinSet_* to coCartesian morphisms in C C^\otimes.

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

Definition

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

Properties

Theorem

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

Corollary

(,1)(\infty,1)-Colimits over simplicial diagrams exists in CMon(C)CMon(C) and are computed in CC if they exist in CC and a preserved by tensor products.

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

Examples

References

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

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