[Haskell-beginners] Type * and * -> *

[Erik Dominikus]

> There's something fundamental I'm missing.

It is not necessarily true, but useful, to think that:

- A *type is* a set of values.
- A *kind* as a set of types.
- The *kind* * is the set of all types.
- The *kind* * -> * is the set of every function with domain * and codomain
*.
- The *type* T -> U is the set of every function with domain T and codomain
U, if each of T and U is a type (a set of values).

For example:

- The type *Bool* is the set {False, True}.
- The type *Maybe Bool* is the set {Nothing, Just False, Just True}.
- *Maybe* is a function such that Maybe(T) = { Nothing } union { Just t | t
in T }.
- *Maybe* is not a type; *Maybe Bool* is a type.
- *a -> a* is the set { \ x -> x }.

Currying and application can happen at both the value level and the type
level:

- The kind of *Either* is ** -> * -> **.
- The kind of *Either a* is ** -> **, if the kind of *a* is ***.
- The kind of *Either a b* is ***, if the kind of *a* is *** and the kind
of *b* is ***.
- The type of *Just* is *a -> Maybe a*.
- The type of *Just x* is *Maybe a*, if the type of *x* is *a*.

> Why is it giving two separate treatments?

It is to show the various ways one *can*
implement/represent/realize/concretize/encode/model
a mathematical construction in Haskell.

You *can* (do type-level programming in Haskell, use all features of C++,
eat spaghetti with a straw, etc.), but the real question has always been:
*should* you?

> If anyone knows of a really thorough and definitive *and *understandable
treatment of Haskell types, I'd appreciate it.

If you mean Haskell 98, then the Haskell 98 Report [1] (especially Chapter
4) seems "thorough and definitive", but I don't know whether you will find
it "understandable".

If you mean the latest Haskell as implemented by GHC, I don't know.

[1] https://www.haskell.org/onlinereport/
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://mail.haskell.org/pipermail/beginners/attachments/20210314/d71b22ff/attachment.html>