nLab horizontal composition

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

• In Cat, horizontal composition is the Godement product of natural transformations.

Related topics

• whiskering