haskell haskell@gentoo.org Provides a wide array of semigroupoids and operations for working with semigroupds. A Semigroupoid is a Category without the requirement of identity arrows for every object in the category. When working with comonads you often have the @\<*\>@ portion of an @Applicative@, but not the @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: > Traversable <---- Foldable <--- Functor ------> Alt ---------> Plus Semigroupoid > | | | | | > v v v v v > Traversable1 <--- Foldable1 Apply --------> Applicative -> Alternative Category > | | | | > v v v v > Bind ---------> Monad -------> MonadPlus Arrow > Apply, Bind, and Extract 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 'Traversable' and 'Foldable' that can be folded with just a 'Semigroup' are added.