nLab
horizontal composition

Contents

Context

2-Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Definition

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

Layer 1 A A α \Downarrow\mathrlap{\alpha} B B β \Downarrow\mathrlap{\beta} C C \mapsto A A β α \Downarrow\beta\circ\alpha C C F 1 F_1 F 2 F_2 G 1 G_1 G 2 G_2 F 2 F 1 F_2\circ F_1 G 2 G 1 G_2\circ G_1
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 AA, BB, and CC then the composition of 1-cells between them defines a functor.

Examples

Last revised on December 7, 2020 at 14:08:08. See the history of this page for a list of all contributions to it.