comonads

Comonads are a concept in category theory that are dual to monads. They consist of an endofunctor W on a category C, together with two natural transformations: a counit ε: W ⇒ Id_C and a comultiplication δ: W ⇒ W ∘ W. These must satisfy the coassociativity and counit laws: δ followed by δ, when composed appropriately, equals δ followed by Wδ (coassociativity), and applying ε after or before δ yields the identity (counit laws). In components, for each object X in C, δ_X: W X → W(W X) and ε_X: W X → X, with the usual naturality conditions.

Intuitively, a comonad models computations that can observe and duplicate context. The comonadic structure provides a way to extract a value from a context and to expand a context to contexts of contexts, enabling context-dependent computations through operations like extend (a higher-level map that threads the comonad’s context through functions).

Common examples arise in concrete categories. The store comonad on Set is a canonical instance: for a set of states S, define W X = Store S X, with extract (Store f s) = f s and duplicate (Store f s) = Store (\s' -> Store f s') s. This captures read-only, indexable stores where the current position is observable. Another familiar example is the stream comonad, where W X is an infinite stream of X values; extract yields the current element, and duplicate yields streams of streams representing all possible continuations.

In functional programming, comonads provide a dual perspective to monads for modeling context, coeffects, and dataflow. They appear in design patterns for context-dependent computations, data streaming, and reversible or zoomable data structures. The theory of comonads parallels that of monads, including notions such as cofree comonads and dual constructions.