nLab
horizontal composition

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 α B β C A α * β C A\qquad\Downarrow\mathrlap{\alpha}\qquad B\qquad\Downarrow\mathrlap{\beta}\qquad C\;\;\mapsto\;\; A\qquad\; \Downarrow\mathrlap{\alpha\ast\beta}\qquad\; C F 1 F_1 F 2 F_2 G 1 G_1 G 2 G_2 F 1 : F 2 F_1\colon F_2 G 1 : G 2 G_1\colon G_2

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

Revised on September 30, 2011 16:21:34 by Tim Porter (193.48.130.2)