\newcommand{&&&}{;;} \newcommand{***}{;|;} \newcommand{+++}{;|;} \newcommand{|||}{;;}

The difficulty of Meijer et al’s classic paper mostly concerns the liberal introduction of new symbols that are hard to parse and remember. In the interest of making it more comprehensible to myself, I translated the bulk of the definitions and laws to their Haskell equivalents, discovering on the way that they have already been codified as the laws for the `Functor`

, `Bifunctor`

, `Profunctor`

, and `Arrow`

classes.

I hope to translate the rest of the paper sometime, but for now here’s section 3.

Note: I later found that Edward Yang had written a similar translation many years prior, albeit not using some of the canonical typeclasses.

As the paper is really dealing with an idealised functional language rooted in category theory, one must keep in mind the convention in category theory to use the same symbol for the action of the functor on both “terms” (objects) and “types” (categories).

Bifunctors and Monofunctors as described in the paper are different names for the typeclasses `Bifunctor`

and `Functor`

, where the type level operation is the higher kinded type itself and the term level operation is the method that characterises the typeclass.

For example, a Bifunctor \(\dagger\) is characterised by its action on both types and terms, where its action on types (in the GADT style) is \(\dagger \;:\;A \rightarrow B \rightarrow A \dagger B\) and on terms \(\dagger \;:\;(A \rightarrow B) \rightarrow(C \rightarrow D) \rightarrow A\dagger C \rightarrow B\dagger D\), satisfying the usual identity and distributivity laws.

In Haskell, \(\dagger\) is a type of kind `* -> * -> *`

with a `bimap`

:

\[\begin{aligned}
id \dagger id &= id \\
f \dagger g \circ h \dagger j &= (f \circ h) \dagger (g \circ j)
\end{aligned}\]

The treatment of monofunctors \(F\) is similar, with the action of \(\cdot_F\) on values given by `fmap`

.

Obviously,

- Product: \(A\*** B = (a,b)\), where the operation on functions is the bimap instance for
`(,)`

- Sum: \(A\+++ B = \text{Either a b}\), where the operation on functions is the bimap instance for
`Either`

.

Something that I never made the connection with before is that if you look at the definitions from the `Control.Arrow`

class, it is clear that

```
(***) :: Arrow a => a b c -> a b' c' -> a (b, b') (c, c')
(+++) :: ArrowChoice a => a b c -> a b' c' -> a (Either b b') (Either c c')
```

are just functors to the product and sum categories.

The various familiar term level functions for products and sums are written as

- \(f\*** g = \texttt{(***)}\)
- \(f \&&& g = \texttt{(&&&)}\)
- \(\grave{\pi}= \mathtt{fst}\)
- \(\acute{\pi}= \mathtt{snd}\)
- \(\grave{\iota}= \mathtt{inl}\)
- \(\acute{\iota}= \mathtt{inr}\)
- \(\Delta x = (x,x) = \texttt{id &&& id}\)
- \(\forall i, \nabla (i,x) = x \cong \texttt{id ||| id}\)

The Arrow functor presented in the paper is in fact the `Profunctor`

instance for `(->)`

in Haskell:

\(\begin{aligned} (f \rightarrow g)\; h = g \circ h \circ f\\ (f \leftarrow_F g)\; h = f \circ h_F \circ g \end{aligned}\)

The second \(\leftarrow_F\) operates on a homomorphically lifted \(h_F\), i.e. `dimap f g . fmap`

.

Of course, `dimap`

distributes over composition:

For monofunctors \(F,G\) and a bifunctor \(\dagger\), the monofunctors \(FG\) and \(F\dagger G\) are defined by

\(\begin{aligned} \tau_{(FG)} &= (\chi_F)_G \\ \tau_{(F\dagger G)} &= \chi_F \dagger \chi_G \end{aligned}\)

where \(\tau\) is a type of kind \(* \rightarrow*\) and \(\chi\) is a function of type \(a \rightarrow b\). (The left hand side is a type, and the right hand side is a term.)

In Haskell:

```
fmap . fmap
:: (Functor f, Functor g)
=> (a -> b) -> (g (f a) -> g (f b))
\f -> bimap (fmap f) (fmap f)
:: (Functor f, Functor g, Bifunctor p)
=> (a -> b) -> (p (f a) (g a) -> p (f b) (g b))
```

Translated literally,

\(\begin{aligned} \grave{\pi}\circ f \| g &= f \circ \grave{\pi}\\ \acute{\pi}\circ f \*** g &= g \circ \acute{\pi}\\ \grave{\pi}\circ f \&&& g &= f \\ \acute{\pi}\circ f \&&& g &= g \\ (\grave{\pi}\circ h) \&&& (\acute{\pi}\circ h) &= h \\ \grave{\pi}\&&& \acute{\pi}&= id \\ f \*** g \circ h \&&& j &= (f \circ h) \&&& (g \circ j) \\ f \&&& g \circ h &= (f \circ h) \&&& (g \circ h) \\ f \*** g = h \*** j &\equiv f = g \land h = j \\ f \&&& g = h \&&& j &\equiv f = g \land h = j \\ f \+++ g \circ \grave{\iota}&= f \\ f \+++ g \circ \acute{\iota}&= g \\ f \||| g \circ \grave{\iota}&= \grave{\iota}\circ f \\ f \||| g \circ \acute{\iota}&= \acute{\iota}\circ g \\ (h \circ \grave{\iota}) \+++ (h \circ \acute{\iota}) &= h && \text {(h strict) } \\ \grave{\iota}\||| \acute{\iota}&= id \\ (f \||| g) \circ (h \+++ j) &= (f \circ h) \||| (g \circ j) \\ f \circ g \||| h &= (f \circ g) \||| (f \circ h) && \text {(f strict) } \\ f \+++ g = h \+++ j &\equiv f = h \land g = j \\ f \||| g = h \||| j &\equiv f = h \land g = j \end{aligned}\)

```
fst . f *** g = f . fst
snd . f *** g = g . snd
fst . f &&& g = f
snd . f &&& g = g
(fst . h) &&& (snd . h) = h
fst &&& snd = id
f *** g . h &&& j = (f . h) &&& (g . j)
f &&& g . h = (f . h) &&& (g . h)
f *** g = h *** j <=> f = g AND h = j
f &&& g = h &&& j <=> f = g AND h = j
f +++ g . Left = f
f +++ g . Right = g
f ||| g . Left = Left . f
f ||| g . Right = Right . g
(h . Left) +++ (h . Right) = h (h strict)
Left ||| Right = id
(f ||| g) . (h +++ j) = (f . h) ||| (g . j)
f . g ||| h = (f . g) ||| (f . h) (f strict)
f +++ g = h +++ j <=> f = h AND g = j
f ||| g = h ||| j <=> f = h AND g = j
```

The Abides law, in three forms:

\(\begin{aligned} (f \&&& g) \||| (h \&&& j) &= (f \||| h) \&&& (g \||| j)\\ \texttt{(f &&& g) ||| (h &&& j)} &= \texttt{(f ||| h) &&& (g ||| j)}\\ \end{aligned}\)

Logically, it is the tautology: \((f \land g) \lor (h \land j) = (f \lor h) \land (g \lor j)\)

It would seem that the arrow combinators `(&&&)`

and `(|||)`

with product/sum types is isomorphic to the boolean semiring.

- \(\mathbf{1} = ()\) (type and constructor have different names a la ML)
- \(void = const ()\)
- \(\mu = fix\)

with laws