Monad (functional Programming) - Comonads

Comonads

Comonads are the categorical dual of monads. They are defined by a type constructor W T and two operations: extract with type W T → T for any T, and extend with type (W T → T' ) → W T → W T' . The operations extend and extract are expected to satisfy these laws:

Alternatively, comonads may be defined in terms of operations fmap, extract and duplicate. The fmap and extract operations define W as a copointed functor. The duplicate operation characterizes comonads: it has type W T → W (W T) and satisfies the following laws:

The two formulations are related as follows:

Whereas monads could be said to represent side-effects, a comonads W represents a kind of context. The extract functions extracts a value from its context, while the extend function may be used to compose a pipeline of "context-dependent functions" of type W A → B.