nLab
algebra in an (infinity,1)-category

Idea

Recall that a monoid object or algebra object in a monoidal category $C$ is the same as a lax monoidal functor

$$*\to C.$$* \to C \,.

This definition generalized to monoidal (∞,1)-categories and defines algebra objects for these.

Definition

For $C$ a monoidal (∞,1)-category with monoidal structure determined by the (∞,1)-functor

$$p_{\otimes }:C^{\otimes }\to N(\Delta {)}^{\mathrm{op}}$$p_\otimes : C^\otimes \to N(\Delta)^{op}

a monoid object of $C$ is a lax monoidal (∞,1)-functor?

$$N(\Delta {)}^{\mathrm{op}}\to C^{\otimes }$$N(\Delta)^{op} \to C^\otimes

Examples

• An algebra object in the stable (infinity,1)-category of spectra is an A-∞ ring. If it is a commutative monoid, it is an E-∞ ring.

Related concepts

• infinity-module

• monoid

• En-algebra

• commutative monoid in an (∞,1)-category

References

definition 1.1.14 in

• Jacob Lurie, Higher Algebra

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

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