问题
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.
回答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.
回答2:
Gabriel Gonzalez has blogged about this. Here's one such post: http://www.haskellforall.com/2012/08/the-category-design-pattern.html
In it, he calls Hask "the function category", and also discusses "the Kleisli category" and "the pipes category." These are all examples of instances of the Category typeclass in Haskell. The Category typeclass in Haskell is a subset of the categories you can find in Haskell.
回答3:
I once uploaded an educational package that demonstrates one example of this. I called it MHask.
http://hackage.haskell.org/package/MHask
Copied from the hackage page:
MHask is the category where
- The objects are Haskell types of kind (* → *) that have an instance of Prelude.Monad
- An arrow from object m to object n is a Haskell function of the form (forall x. m x → n x)
- Arrow composition is merely a specialization of Haskell function composition
- The identity arrow for the object m is the Prelude.id function in Haskell, specialized to (forall x. m x → m x)
Caveat emptor; I have not looked at this in a long time. There may be mistakes.
来源：https://stackoverflow.com/questions/50735311/categorical-structure-in-haskell