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

Idea

The notion of monoid (or monoid object, algebra, algebra object) in a monoidal (infinity,1)-category $C$ is the (infinity,1)-categorical generalization of monoid in a monoidal category.

Definition

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

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

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

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

This generalizes how, for monoidal categories, monoid objects are the same as lax monoidal functors

$$* \to C \,.$$

A generalization to the case that the monoidal (∞,1)-category $C$ is not Cartesian is discussed in Section 4.1.3 of Higher Algebra.

Examples

• A monoid 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 a symmetric monoidal (∞,1)-category

• category object 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 $(\infty,1)$-Categories (arXiv)