Module documentation for 4.3
Provides a wide array of (semi)groupoids and operations for working with them.
Semigroupoid is a
Category without the requirement of identity arrows for every object in the category.
Category is any
Semigroupoid for which the Yoneda lemma holds.
When working with comonads you often have the
<*> portion of an
pure. This was captured in Uustalu and Vene's "Essence of Dataflow Programming"
in the form of the
ComonadZip class in the days before
Applicative. Apply provides a weaker invariant, but for the comonads used for data flow programming (found in the streams package), this invariant is preserved. Applicative function composition forms a semigroupoid.
Similarly many structures are nearly a comonad, but not quite, for instance lists provide a reasonable
extend operation in the form of
tails, but do not always contain a value.
Ideally the following relationships would hold:
Foldable ----> Traversable <--- Functor ------> Alt ---------> Plus Semigroupoid | | | | | v v v v v Foldable1 ---> Traversable1 Apply --------> Applicative -> Alternative Category | | | | v v v v Bind ---------> Monad -------> MonadPlus Arrow
Apply, Bind, and Extend (not shown) give rise the Static, Kleisli and Cokleisli semigroupoids respectively.
This lets us remove many of the restrictions from various monad transformers
as in many cases the binding operation or
<*> operation does not require them.
Finally, to work with these weaker structures it is beneficial to have containers
that can provide stronger guarantees about their contents, so versions of
Foldable that can be folded with just a
Semigroup are added.