# nLab symmetric monoidal groupoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

categorification

## Examples

#### Category theory

monoid theory in algebra:

# Contents

## Idea

A braided monoidal groupoid is a braided monoidal category whose underlying category happens to be a groupoid (hence all whose morphisms are isomorphisms).

Equivalently: A braided monoidal groupoid whose dagger adjoint of the braiding is the opposite braiding,

## Definitions

A symmetric monoidal groupoid is a braided monoidal groupoid $G$ such that for all objects $A$ and $B$, the dagger adjoint or 2-sided inverse of the braiding $\beta_{A,B} : A \otimes B \cong^\dagger B \otimes A$ is the braiding $\beta_{B,A} : B \otimes A \cong^\dagger A \otimes B$

$\beta_{A,B}^\dagger = \beta_{B,A}$