 Boost.Hana  1.7.1 Your standard library for metaprogramming 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

## 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