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

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

• Saunders MacLane, §II.5, p. 43 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]