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

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

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

(∞,1)-category theory

# Contents

## 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 monoidal (∞,1)-category with 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

$* \to C \,.$

## References

definition 1.1.14 in

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

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

Last revised on May 13, 2019 at 16:56:53. See the history of this page for a list of all contributions to it.