# semigroupoids-5.2.1: 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.

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 `Traversable`

and `Foldable`

that can be folded with just a `Semigroup`

are added.

- Data
- Bifunctor
- Functor
- Data.Groupoid
- Data.Isomorphism
- Semigroup
- Data.Semigroupoid
- Traversable