# comonad

This package provides comonads, the categorical dual of monads. The typeclass
provides three methods: `extract`

, `duplicate`

, and `extend`

.

```
class Functor w => Comonad w where
extract :: w a -> a
duplicate :: w a -> w (w a)
extend :: (w a -> b) -> w a -> w b
```

There are two ways to define a comonad:

I. Provide definitions for `extract`

and `extend`

satisfying these laws:

```
extend extract = id
extract . extend f = f
extend f . extend g = extend (f . extend g)
```

In this case, you may simply set `fmap`

= `liftW`

.

These laws are directly analogous to the laws for
monads. The comonad laws can
perhaps be made clearer by viewing them as stating that Cokleisli composition
must be a) associative and b) have `extract`

for a unit:

```
f =>= extract = f
extract =>= f = f
(f =>= g) =>= h = f =>= (g =>= h)
```

II. Alternately, you may choose to provide definitions for `fmap`

,
`extract`

, and `duplicate`

satisfying these laws:

```
extract . duplicate = id
fmap extract . duplicate = id
duplicate . duplicate = fmap duplicate . duplicate
```

In this case, you may not rely on the ability to define `fmap`

in
terms of `liftW`

.

You may, of course, choose to define both `duplicate`

/and/ `extend`

.
In that case, you must also satisfy these laws:

```
extend f = fmap f . duplicate
duplicate = extend id
fmap f = extend (f . extract)
```

These implementations are the default definitions of `extend`

and`duplicate`

and
the definition of `liftW`

respectively.

Contributions and bug reports are welcome!

Please feel free to contact me through github or on the #haskell IRC channel on irc.freenode.net.

-Edward Kmett