# nLab 2-morphism

Contents

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Definition

A 2-morphism in an n-category is a k-morphism for $k = 2$: it is a higher morphism between ordinary 1-morphisms.

So in the hierarchy of $n$-categories, the first step where 2-morphisms appear is in a 2-category. This includes cases such as bicategory, 2-groupoid or double category.

## Shapes

There are different geometric shapes for higher structures: globes, simplices, cubes, etc. Accordingly, 2-morphisms may appear in different guises:

A globular $2$-morphism looks like this:

$a\mathrlap{\begin{matrix}\begin{svg} \end{svg}\includegraphics[width=56]{curvearrows}\end{matrix}}{\phantom{a}\space{0}{0}{13}\Downarrow\space{0}{0}{13}\phantom{a}} b$

A simplicial $2$-morphism looks like this:

$\begin{matrix} && b \\ & \nearrow &\Downarrow& \searrow \\ a &&\to&& c \end{matrix}$

A cubical $2$-morphism looks like this:

$\begin{matrix} & & b \\ & \nearrow & & \searrow \\ a & & \Downarrow & & d \\ & \searrow & & \nearrow \\ & & c \end{matrix}$

Of course, using identity morphisms and composition, we can turn one into the other; which is more fundamental depends on which shapes you prefer.

## Examples

Last revised on November 2, 2022 at 09:20:04. See the history of this page for a list of all contributions to it.