Semigroupoids: Category sans id
Provides a wide array of (semi)groupoids and operations for working
with them.
A
Semigroupoid is a
Category without the requirement of
identity arrows for every object in the category.
A
Category is any
Semigroupoid for which the Yoneda
lemma holds.
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.
We describe the relationships between the type classes defined in this
package and those from
base (and some from
contravariant) in the diagram below. Thick-bordered nodes
correspond to type classes defined in this package; thin-bordered ones
correspond to type classes from elsewhere. Solid edges indicate a
subclass relationship that actually exists; dashed edges indicate a
subclass relationship that
should exist, but currently doesn't.
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
Traversable and
Foldable that can be
folded with just a
Semigroup are added.