# nLab horizontal composition

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

#### Higher category theory

higher category theory

# Contents

## Definition

In a 2-category, the composition of 2-morphisms along objects is called horizontal composition .

category: svg

This is in contrast to the vertical composition of 2-morphisms, which is their composition along 1-morphisms.

## Properties

Horizontal and vertical composition are subject to the compatibility condition called the interchange law, which in this case and the higher categorical ones means that if we fix three objects $A$, $B$, and $C$ then the composition of 1-cells between them defines a functor.

## Examples

Last revised on December 7, 2020 at 19:08:08. See the history of this page for a list of all contributions to it.