Comonad

The Comonad concept represents context-sensitive computations and data.

Formally, the Comonad concept is dual to the Monad concept. But unless you're a mathematician, you don't care about that and it's fine. So intuitively, a Comonad represents context sensitive values and computations. First, Comonads make it possible to extract context-sensitive values from their context with extract. In contrast, Monads make it possible to wrap raw values into a given context with lift (from Applicative).

Secondly, Comonads make it possible to apply context-sensitive values to functions accepting those, and to return the result as a context-sensitive value using extend. In contrast, Monads make it possible to apply a monadic value to a function accepting a normal value and returning a monadic value, and to return the result as a monadic value (with chain).

Finally, Comonads make it possible to wrap a context-sensitive value into an extra layer of context using duplicate, while Monads make it possible to take a value with an extra layer of context and to strip it with flatten.

Whereas lift, chain and flatten from Applicative and Monad have signatures

\begin{align*} \mathtt{lift}_M &: T \to M(T) \\ \mathtt{chain} &: M(T) \times (T \to M(U)) \to M(U) \\ \mathtt{flatten} &: M(M(T)) \to M(T) \end{align*}

extract, extend and duplicate from Comonad have signatures

\begin{align*} \mathtt{extract} &: W(T) \to T \\ \mathtt{extend} &: W(T) \times (W(T) \to U) \to W(U) \\ \mathtt{duplicate} &: W(T) \to W(W(T)) \end{align*}

Notice how the "arrows" are reversed. This symmetry is essentially what we mean by Comonad being the dual of Monad.

Note

The Typeclassopedia is a nice Haskell-oriented resource for further reading about Comonads.

Minimal complete definition

extract and (extend or duplicate) satisfying the laws below. A Comonad must also be a Functor.

Laws

For all Comonads w, the following laws must be satisfied:

extract(duplicate(w)) == w
transform(duplicate(w), extract) == w
duplicate(duplicate(w)) == transform(duplicate(w), duplicate)

Note

There are several equivalent ways of defining Comonads, and this one is just one that was picked arbitrarily for simplicity.

Refined concept

1. Functor
   Every Comonad is also required to be a Functor. At first, one might think that it should instead be some imaginary concept CoFunctor. However, it turns out that a CoFunctor is the same as a Functor, hence the requirement that a Comonad also is a Functor.

Concrete models

hana::lazy