cartesian package:constrained-categories

Quite a few categories (monoidal categories) will permit "products" of objects as objects again – in the Haskell sense those are tuples – allowing for "dyadic morphisms" (x,y) ~> r. Together with a unique UnitObject, this makes for a monoidal structure, with a few natural isomorphisms. Ordinary tuples may not always be powerful enough to express the product objects; we avoid making a dedicated associated type for the sake of simplicity, but allow for an extra constraint to be imposed on objects prior to consideration of pair-building. The name Cartesian is disputable: in category theory that would rather Imply cartesian closed category (which we represent with Curry). Monoidal would make sense, but we reserve that to Functors.

Monoidal categories need not be based on a cartesian product. The relevant alternative is coproducts. The dual notion to Cartesian replaces such products (pairs) with sums (Either), and unit () with void types. Basically, the only thing that doesn't mirror Cartesian here is that we don't require CoMonoid (ZeroObject k). Comonoids do in principle make sense, but not from a Haskell viewpoint (every type is trivially a comonoid). Haskell of course uses sum types, variants, most often without Either appearing. But variants are generally isomorphic to sums; the most important (sums of unit) are methods here.