Categorical structure in Haskell

Question

Hask is usually thought to be the category whose objects are types and morphisms are functions. However, I've seen Conor McBride (@pigworker) warn against the use of Hask multiple times (1, 2, 3):

• I would discourage talk of "the Hask Category" because it subconsciously conditions you against looking for other categorical structure in Haskell programming.

• Note, I dislike the use of "Hask" as the name of the "category of Haskell types and functions": I fear that labelling one category as the Haskell category has the unfortunate side-effect of blinding us to the wealth of other categorical structure in Haskell programming. It's a trap.

• I wish people wouldn't call it "Hask", though: it threatens to limit the imagination.

What other categories can we see in Haskell?

In one of his answers, he touches upon some of these ideas, but I wonder if someone could expand upon it; and I wonder if there are even more examples.

[...] there's a ton of categorical structure lurking everywhere, there's certainly a ton of categorical structure available (possibly but not necessarily) at higher kinds. I'm particularly fond of functors between indexed families of sets.

Answer 1

Constraints in Haskell also form a category. The objects are the constraints, and the arrows mean "this constraint implies this other constraint". So every constraint implies itself, and there's an arrow between Monad f and Applicative f, between Ord a and Eq a and between Ord a and Ord [a].

It is a thin category, so there is at most one arrow between two objects.