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

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Contents

Idea

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

Definition

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

p ⊗:C ⊗→N(Δ) op p_\otimes : C^\otimes \to N(\Delta)^{op}

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

N(Δ) op→C ⊗ N(\Delta)^{op} \to C^\otimes

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

*→C. * \to C \,.

A generalization to the case that the monoidal (∞,1)-category CC 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.

• 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 (∞,1)(\infty,1)-Categories (arXiv)