# nLab 2-morphism

Contents

### Context

#### 2-Category theory

2-category theory

# 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.

Eric: Are there any consistency requirements for a 2-morphism? For example, in the bigon above, if $f:a\to b$, $g:a\to b$, and $\alpha:f\to g$, are there requirements on $\alpha:f\to g$ regarding $f$ and $g$? For example, should $\alpha$ come with component 1-morphisms $\alpha_a:a\to a$ and $\alpha_b:b\to b$ such that

$\alpha_a\circ g = f\circ\alpha_b$

or maybe

$\alpha_a\circ g \simeq f\circ\alpha_b$

? Could there be a 2-morphism without the corresponding 1-morphism components?

Urs Schreiber: in any given 2-category you have to specify which 2-morphisms exactly there are supposed to be, what $\alpha$ exactly you allow between $f$ and $g$. When you ask about components, it seems you are thinking of 2-morphisms specifically in the 2-category Cat. Here, yes, the allowed 2-morphisms are those that are natural transformations between their source and target 1-morphisms, which are functors.

Eric: I think the exchange law might be what I had in mind.

## Examples

Last revised on May 25, 2016 at 14:11:41. See the history of this page for a list of all contributions to it.