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



Higher algebra

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

Monoid theory



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.


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) opC 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.



definition 1.1.14 in

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

Last revised on August 24, 2023 at 09:19:13. See the history of this page for a list of all contributions to it.