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commutative algebra in an (infinity,1)-category

Idea

Recall the pattern in ordinary algebra:

• In a monoidal category one can consider monoid or algebra objects.

• In a symmetric monoidal category one can consider commutative monoid or algebra objects.

Accordingly in higher algebra:

• In a monoidal (infinity,1)-category one can consider algebra in an (infinity,1)-category.

• In a symmetric monoidal (infinity,1)-category one can consider commutative algebra objects.

Definition

A commutative algebra object in a symmetric monoidal (infinity,1)-category $C$ is a lax symmetric monoidal $(\mathrm{\infty },1)$-functor

$$*\to C.$$* \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

$$p:C^{\otimes }\to N({\mathrm{FinSet}}_{*})$$p : C^\otimes \to N(FinSet_*)

a commutative algebra object in $C$ is a section

$$A:N({\mathrm{FinSet}}_{*})\to C^{\otimes }$$A : N(FinSet_*) \to C^\otimes

such that $A$ carries collapsing morphisms in ${\mathrm{FinSet}}_{*}$ to coCartesian morphisms in $C^{\otimes }$.

Examles

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

References

the above definition is definition 1.19 in

• Jacob Lurie, Commutative algebra