nLab vertical composition

Definition

In a 2-category (and more generally in higher category theory) 2-morphisms have a composition along the 1-morphisms that they go between.

$$\array{ & \nearrow &\Downarrow& \searrow \\ x &&\to&& y \\ & \searrow &\Downarrow& \nearrow } \,.$$

This is in contrast to the other composition operation, along objects, which is called horizontal composition.

Examples

• In the 2-category Cat, 2-morphisms are natural transformations and vertical composition is the composition of these, the composition in the corresponding functor category.

• For $A$ an abelian group and $\mathbf{B}^2 A$ its double delooping 2-groupoid, vertical composition (as well as horizontal composition) is given by the group product operation in $A$.

References

For the notion of vertical composition of natural transformations (ie. composition in functor categories):

• Saunders MacLane, §II.4 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]