GoKawiilPreviously: Profunctor Equipment.
To make things more palatable for programmers, I decided to provide a toy implementation of some of the equipments in Haskell. The advantage of this encoding is that it can be verified by the compiler, and I still trust the compiler more than I trust the AI.
A more adequate implementation would require a full-blown dependently typed language, but if we restrict ourselves to just a single category and work only with endo-functors and endo-profunctors, we can get at least some intuitions. If you want to see a more elaborate version, see the proarrows library by Sjoerd Visscher.
The only 0-cell I’ll be using is the Haskell category of types and functions. For vertical 1-cells I’ll use the standard library implementation of Functor , and for horizontal ones I’ll use Profunctor .
A 2-cell in :
is implemented as a natural transformation:
type Cell f g h j = forall a c . h a c -> j (f a) (g c)
The forall serves as a universal quantifier.
The horizontal composition of such cells is given by:
hcomp :: (Functor f, Functor f', Functor g, Functor g'
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