• object

• morphism

• 2-morphism

• 3-morphism

• k-morphism

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 simplicial $2$-morphism looks like this:

A cubical $2$-morphism looks like this:

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

• In the 2-category Cat, 2-morphisms are natural transformations between functors.

• In a path 2-groupoid 2-morphisms are certain surfaces or images of surfaces in a space, going between paths in that space.