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.

Examples

Last revised on May 10, 2018 at 09:11:17. See the history of this page for a list of all contributions to it.