Co-Yoneda Lemma

by oleksandrmanzyuk

The purpose of this humble blog post is to understand this insightful comment by Edward Kmett to the question “Is operational really isomorphic to a free monad?” posted at StackOverflow.

Consider the following data type definition in Haskell, which can also be found in Edward Kmett’s package kan-extensions:

data CoYoneda f a = forall b. CoYoneda (b -> a) (f b)

In Haskell, a forall before the data constructor means existential rather than universal quantification, so that CoYoneda f a is isomorphic to the type \exists b. (b\to a, f\, b). The forall hints at the polymorphic type of the constructor

CoYoneda :: forall b. (b -> a) -> f b -> CoYoneda f a

Every polymorphic function satisfies a theorem that can be extracted by reading its type as a relation. This is referred to as parametricity. The reader is referred to Wadler’s classic paper “Theorems for free!” for more details.

Suppose that f is a functor. Then parametricity implies the following free theorem for the function CoYoneda: if g :: b -> a and y :: f c, then CoYoneda (g . h) y == CoYoneda g (fmap h y) for any h :: c -> b.

Consider the following two functions:

phi :: f a -> CoYoneda f a
phi x = CoYoneda id x

psi :: Functor f => CoYoneda f a -> f a
psi (CoYoneda g y) = fmap g y

Then obviously psi . phi == id:

(psi . phi) x == fmap id x == x

The above free theorem implies also that phi . psi == id:

(phi . psi) (CoYoneda g y) == CoYoneda id (fmap g y) == CoYoneda g y

We conclude that CoYoneda f is isomorphic f. This is a Haskell version of the co-Yoneda Lemma.

Appendix

The only thing that is missing in Wadler’s paper is how to associate a relation with the type f b. Wadler only explains this in the case of the list functor. We assume that the Haskell functor f is modeled by a functor F on the category of sets. Suppose that \mathcal{B}: B \Leftrightarrow B' is a relation associated with the type b. That is, \mathcal{B} is a subset of B\times B'. Let \iota: \mathcal{B}\hookrightarrow B\times B' denote the inclusion map. We define the relation associated with the type f b as the image of the composite \langle F\pi, F\pi'\rangle\circ F\iota: F(\mathcal{B})\to FB\times FB', where \pi: B\times B'\to B and \pi': B\times B'\to B' are the canonical projections.

For example, if f = [], so that F=(-)^* is the list functor, then the above definition says that two lists xs \in B^* and ys \in B'^* are related if there exists a list zs\in \mathcal{B}^* \subset (B\times B')^* such that

\displaystyle  \begin{array}{rcl} map\; fst\; zs & \equiv & xs,\\ map\; snd\; zs & \equiv & ys. \end{array}

In other words, xs and ys are related if they have the same length and corresponding elements are related, which coincides with Wadler’s definition.

It follows from this definition that if \mathcal{B} is the relation associated with a function g: B\to B', then F(\mathcal{B}) is the relation associated with the function Fg. Indeed, if \mathcal{B} is the graph of g, i.e., the image of the map \langle\mathrm{id}_B, g\rangle: B\to B\times B', then F(\mathcal{B}) is the image of the composite \langle F\pi, F\pi'\rangle\circ F(\langle\mathrm{id}_B, g\rangle), which coincides with \langle F\mathrm{id}_B, Fg\rangle = \langle \mathrm{id}_{FB}, Fg\rangle. Therefore, F(\mathcal{B}) coincides with the graph of Fg.

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