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

Context

Higher algebra

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

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 monoidal (∞,1)-category with 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 \,.

Examples

References

definition 1.1.14 in

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

Revised on March 9, 2015 12:16:02 by Adeel Khan (93.131.147.91)