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

References

Horizontal composition of natural transformations is first describes in

  • Roger Godement, Appendix (pp. 269) of: Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

for textbook accounts see most of those listed at category, such as

Last revised on August 18, 2023 at 15:20:12. See the history of this page for a list of all contributions to it.