CODEWITHSUNDEEP
functorcategory-theory
Ryba
Ryba

Reputation: 701

Confusion in understanding horisontal composition of natural transformations

I'm currently reading Category Theory for Programmers by Bartosz Milewski. In chapter about natural transformation i found a following paragraph:

Let’s focus on two objects of π‚πšπ­ β€” categories 𝐂 and 𝐃. There is a set of natural transformations that go between functors that connect 𝐂 to 𝐃. These natural transformations are our new arrows from 𝐂 to 𝐃. By the same token, there are natural transformations going between functors that connect 𝐃 to 𝐄, which we can treat as new arrows going from 𝐃 to 𝐄. Horizontal composition is the composition of these arrows.

Could someone explain how did natural trasformation between functors that connect categories C and D suddenly became arrows connecting C and D, which would make it a functor again?

Upvotes: 1

Views: 76

Answers (1)

Bartosz Milewski
Bartosz Milewski

Reputation: 11650

It so happens that Cat has more structure than the obvious one: a category with categories as objects and functors as morphisms. There is a second layer, in which we combine all natural transformations between all pairs of functors from C to D and call it a hom-set from C to D. We define composition between these hom-sets in terms of horizontal composition of natural transformations. Notice, these new hom-sets are different from old hom-sets, which are sets of functors.

Upvotes: 1

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