A Haskell library for doing category theory with a central role for profunctors.
Kind-indexed categories makes life a lot easier, once you know what the kind is of a type, you know which category it belongs to.
Using kind-indexed categories means you cannot share objects between categories. Newtype
wrappers fix this. For example, if you have a category for kind k, it's opposite category
has kind OP k.
If profunctors would have kind OP j -> k -> Type, then (->) wouldn't be a profunctor
as is. This would require too many wrapper all over the place. So instead j -> k -> Type
is reserved for profunctors. This means that bifunctors need to use (j, k) -> Type.
You need this already when creating a category of functors, then each object needs a
Functor constraint. It turns out this is powerful enough to limit the objects of any
type of category.
If you're not careful these objects constraints can become unweildy, requiring a long list
of object constraints for each function. But if you have an arrow from a to b, that's
proof enough that a and b are objects. So there are functions (//) and (\\) to
observe the constraints.
Functors have kind j -> k, but you can't just make a datatype of any kind, it must
always be of the shape j -> k -> ... -> Type. So for example you can't make an
identity functor that works for any k. But functors are isomorphic to representable
profunctors, with kind k -> j -> Type. (Note that the kinds swap!) So you can write
an identity representable profunctor!
To make working with representable profunctors instead of functors, the category theory should work with profunctors where possible.