# nLab monoidal groupoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

categorification

category theory

# Contents

## Idea

A monoidal category whose underlying category is a groupoid (hence all whose morphisms are invertible).

Equivalently: an $A_4$-spatial groupoid whose unitors satisfy the triangle identities;

## Definitions

A monoidal groupoid is an $A_4$-spatial groupoid $G$ such that the triangle identity is satisfied for all objects $A:G$ and $B:G$:

$\tau_{A,B}:(id_A \otimes \lambda_B) \circ \alpha_{A,I,B} = (\rho_A \otimes id_B)$

## Properties

The oidification of a monoidal groupoid is a (2,1)-category.