A Notation for Comonads

Abstract

The category-theoretic concept of a monad occurs widely as a design pattern for functional programming with effects. The utility and ubiquity of monads is such that some languages provide syntactic sugar for this pattern, further encouraging its use. We argue that comonads, the dual of monads, similarly provide a useful design pattern, capturing notions of context dependence. However, comonads remain relatively under-used compared to monads—due to a lack of knowledge of the design pattern along with the lack of accompanying simplifying syntax.

We propose a lightweight syntax for comonads in Haskell, analogous to the do-notation for monads, and provide examples of its use. Via our notation, we also provide a tutorial on programming with comonads.

A Proof of Shape Preservation

To prove shape preservation we first prove the following intermediate lemma:

$$\begin{aligned} cmap \; g \mathbin {\circ }extend \; f \mathrel {=}extend \;( g \mathbin {\circ } f ) \end{aligned}$$

(7)

$$\begin{aligned} \begin{array}{rll} &{} cmap \; g \mathbin {\circ }extend \; f &{} \\ \equiv \;&{} extend \;( g \mathbin {\circ }extract )\mathbin {\circ }extend \; f &{} \quad \text {definition of cmap } \\ \equiv \;&{} extend \;( g \mathbin {\circ }extract \mathbin {\circ }extend \; f ) &{} \quad \text {[C3]} \\ \equiv \;&{} extend \;( g \mathbin {\circ } f ) \quad \Box &{} \quad \text {[C2]} \end{array} \end{aligned}$$

The proof of shape preservation (3) is then:

$$\begin{aligned} \begin{array}{rll} &{} shape \mathbin {\circ }(extend \; f ) &{} \\ \equiv \;&{} ( cmap \;( const \;())\mathbin {\circ }(extend \; f ) &{} \text {definition of shape } \\ \equiv \;&{} extend \;(( const \;())\mathbin {\circ } f ) &{} (7) \\ \equiv \;&{} extend \;(( const \;())\mathbin {\circ }extract ) &{} ( const \; x )\mathbin {\circ } f \equiv ( const \; x )\mathbin {\circ } g \\ \equiv \;&{} ( cmap \;( const \;()))\mathbin {\circ }(extend \;extract ) \quad \quad &{} (7) \\ \equiv \;&{} cmap \;( const \;()) &{} \text {[C1]} \\ \equiv \;&{} shape &{} \text {definition of shape } \end{array} \end{aligned}$$

$\Box$