# nLab vertical composition

### Context

#### 2-Category theory

2-category theory

## Structures on 2-categories

#### Higher category theory

higher category theory

# Contents

## Definition

In a 2-category (and more generally in higher category theory) 2-morphisms have a composition along the 1-morphisms that they go between.

$\array{ & \nearrow &\Downarrow& \searrow \\ x &&\to&& y \\ & \searrow &\Downarrow& \nearrow } \,.$

This is in contrast to the other composition operation, along objects, which is called horizontal composition.

## Examples

• In the 2-category Cat, 2-morphisms are natural transformations and vertical composition is the composition of these, the composition in the corresponding functor category.

• For $A$ an abelian group and $\mathbf{B}^2 A$ its double delooping 2-groupoid, vertical composition (as well as horizontal composition) is given by the group product operation in $A$.

Created on September 9, 2010 08:12:37 by Urs Schreiber (77.80.22.58)