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

Contents

Context

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

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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

Last revised on March 9, 2015 at 12:35:40. See the history of this page for a list of all contributions to it.