In this post, I’m going to introduce a new class of combinators for pipes, with an interesting categorical interpretation. I will be using the pipe implementation of my previous post.

```
> {-# LANGUAGE MultiParamTypeClasses #-}
> {-# LANGUAGE FlexibleInstances #-}
> {-# LANGUAGE TypeFamilies #-}
> {-# LANGUAGE GeneralizedNewtypeDeriving #-}
> module Blog.Pipes.MonoidalInstances where
>
> import Blog.Pipes.Guarded hiding (groupBy)
> import qualified Control.Arrow as A
> import Control.Category
> import Control.Categorical.Bifunctor
> import Control.Category.Associative
> import Control.Category.Braided
> import Control.Category.Monoidal
> import Control.Monad (forever)
> import Control.Monad.Free
> import Data.Maybe
> import Data.Void
> import Prelude hiding ((.), id, filter, until)
```

When pipes were first released, some people noticed the lack of an `Arrow`

instance. In fact, it is not hard to show that, even identifying pipes modulo some sort of observational equality, there is no `Arrow`

instance that satisfies the arrow laws.

The problem, of course, is with `first`

, because we already have a simple implementation of `arr`

. If we try to implement `first`

we immediately discover that there's a problem with the `Yield`

case:

`first (Yield x c) = yield (x, ???) >> first c`

Since ??? can be of any type, the only possible value is bottom, which of course we don't want to introduce. Alternative definitions of `first`

that alter the structure of a yielding pipe are not possible if we want to satisfy the law:

`first p >+> pipe fst == pipe fst >+> p`

Concretely, the problem is that the cartesian product in the type of `first`

forces a sort of "synchronization point" that doesn't necessarily exist. This is better understood if we look at the type of `(***)`

, of which `first`

can be thought of as a special case:

```
(***) :: Arrow k => k a b -> k a' b' -> k (a, a') (b, b')
first = (*** id)
```

If the two input pipes yield at different times, there is no way to faithfully match their yielded values into a pair. There are hacks around that, but they don't behave well compositionally, and exhibit either arbitrarily large space leaks or data loss.

This has been addressed before: stream processors, like those of the Fudgets library, being very similar to Pipes, have the same problem, and some resolutions have been proposed, although not entirely satisfactory.

## Arrows as monoidal categories

It is well known within the Haskell community that Arrows correspond to so called Freyd categories, i.e. premonoidal categories with some extra structures.

Using the `Monoidal`

class by Edward Kmett (now in the `categories`

package on Hackage), we can try to make this idea precise.

Unfortunately, we have to use a newtype to avoid overlapping instances in the case of the Hask category:

```
> newtype ACat a b c = ACat { unACat :: a b c }
> deriving (Category, A.Arrow)
```

First, cartesian products are a bifunctor in the category determined by an Arrow.

```
> instance A.Arrow a => PFunctor (,) (ACat a) (ACat a) where
> first = ACat . A.first . unACat
> instance A.Arrow a => QFunctor (,) (ACat a) (ACat a) where
> second = ACat . A.second . unACat
> instance A.Arrow a
> => Bifunctor (,) (ACat a) (ACat a) (ACat a) where
> bimap (ACat f) (ACat g) = ACat $ f A.*** g
```

Now we can say that products are associative, using the associativity of products in Hask:

```
> instance A.Arrow a => Associative (ACat a) (,) where
> associate = ACat $ A.arr associate
> instance A.Arrow a => Disassociative (ACat a) (,) where
> disassociate = ACat $ A.arr disassociate
```

Where we use the `Disassociative`

instance to express the inverse of the associator. And finally, the Monoidal instance:

```
> type instance Id (ACat a) (,) = ()
> instance A.Arrow a => Monoidal (ACat a) (,) where
> idl = ACat $ A.arr idl
> idr = ACat $ A.arr idr
> instance A.Arrow a => Comonoidal (ACat a) (,) where
> coidl = ACat $ A.arr coidl
> coidr = ACat $ A.arr coidr
```

Where, again, the duals are actually inverses. Also, products are symmetric:

```
> instance A.Arrow a => Braided (ACat a) (,) where
> braid = ACat $ A.arr braid
> instance A.Arrow a => Symmetric (ACat a) (,)
```

As you see, everything is trivially induced by the cartesian structure on Hask, since `A.arr`

gives us an identity-on-objects functor. Note, however, that the `Bifunctor`

instance is legitimate only if we assume a strong commutativity law for arrows:

`first f >>> second g == second g >>> first f`

which we will, for the sake of simplicity.

## Replacing products with arbitrary monoidal structures

Once we express the Arrow concept in terms of monoidal categories, it is easy to generalize it to arbitrary monoidal structures on Hask.

In particular, coproducts work particularly well in the category of pipes:

```
> instance Monad m
> => PFunctor Either (PipeC m r) (PipeC m r) where
> first = PipeC . firstP . unPipeC
>
> firstP :: Monad m => Pipe a b m r
> -> Pipe (Either a c) (Either b c) m r
> firstP (Pure r) = return r
> firstP (Free (M m)) = lift m >>= firstP
```

Yielding a sum is now easy: just yield on the left component.

`> firstP (Free (Yield x c)) = yield (Left x) >> firstP c`

Awaiting is a little bit more involved, but still easy enough: receive left and null values normally, and act like an identity on the right.

```
> firstP (Free (Await k)) = go
> where
> go = tryAwait
> >>= maybe (firstP $ k Nothing)
> (either (firstP . k . Just)
> (\x -> yield (Right x) >> go))
```

And of course we have an analogous instance on the right:

```
> instance Monad m
> => QFunctor Either (PipeC m r) (PipeC m r) where
> second = PipeC . secondP . unPipeC
>
> secondP :: Monad m => Pipe a b m r
> -> Pipe (Either c a) (Either c b) m r
> secondP (Pure r) = return r
> secondP (Free (M m)) = lift m >>= secondP
> secondP (Free (Yield x c)) = yield (Right x) >> secondP c
> secondP (Free (Await k)) = go
> where
> go = tryAwait
> >>= maybe (secondP $ k Nothing)
> (either (\x -> yield (Left x) >> go)
> (secondP . k . Just))
```

And a bifunctor instance obtained by composing `first`

and `second`

in arbitrary order:

```
> instance Monad m
> => Bifunctor Either (PipeC m r)
> (PipeC m r) (PipeC m r) where
> bimap f g = first f >>> second g
```

At this point we can go ahead and define the remaining instances in terms of the identity-on-objects functor given by `pipe`

:

```
> instance Monad m => Associative (PipeC m r) Either where
> associate = PipeC $ pipe associate
> instance Monad m => Disassociative (PipeC m r) Either where
> disassociate = PipeC $ pipe disassociate
>
> type instance Id (PipeC m r) Either = Void
> instance Monad m => Monoidal (PipeC m r) Either where
> idl = PipeC $ pipe idl
> idr = PipeC $ pipe idr
> instance Monad m => Comonoidal (PipeC m r) Either where
> coidl = PipeC $ pipe coidl
> coidr = PipeC $ pipe coidr
>
> instance Monad m => Braided (PipeC m r) Either where
> braid = PipeC $ pipe braid
> instance Monad m => Symmetric (PipeC m r) Either
```

## Multiplicative structures

There is still a little bit of extra structure that we might want to exploit. Since `PipeC m r`

is a monoidal category, it induces a (pointwise) monoidal structure on its endofunctor category, so we can speak of monoid objects there. In particular, if the identity functor is a monoid, it means that we can define a "uniform" monoid structure for all the objects of our category, given in terms of natural transformations (i.e. polymorphic functions).

We can represent this specialized monoid structure with a type class (using kind polymorphism and appropriately generalized category-related type classes, it should be possible to unify this class with `Monoid`

and even `Monad`

, similarly to how it's done here):

```
> class Monoidal k p => Multiplicative k p where
> unit :: k (Id k p) a
> mult :: k (p a a) a
```

Dually, we can have a sort of uniform coalgebra:

```
> class Comonoidal k p => Comultiplicative k p where
> counit :: k a (Id k p)
> comult :: k a (p a a)
```

The laws for those type classes are just the usual laws for a monoid in a (not necessarily strict) monoidal category:

```
first unit . mult == idl
second unit . mult == idr
mult . first mult == mult . second mult . associate
first counit . comult == coidl
second counit . comult == coidr
first diag . diag == disassociate . second diag . diag
```

Now, products have a comultiplicative structure on Hask (as in every category with finite products), given by the terminal object and diagonal natural transformation:

```
> instance Comultiplicative (->) (,) where
> counit = const ()
> comult x = (x, x)
```

while coproducts have a multiplicative structure:

```
> instance Multiplicative (->) Either where
> unit = absurd
> mult = either id id
```

that we can readily transport to `PipeC m r`

using `pipe`

:

```
> instance Monad m => Multiplicative (PipeC m r) Either where
> unit = PipeC $ pipe absurd
> mult = PipeC $ pipe mult
```

Somewhat surprisingly, pipes also have a comultiplicative structure of their own:

```
> instance Monad m => Comultiplicative (PipeC m r) Either where
> counit = PipeC discard
> comult = PipeC . forever $ do
> x <- await
> yield (Left x)
> yield (Right x)
```

## Heterogeneous metaprogramming

All the combinators we defined can actually be used in practice, and the division in type classes certainly sheds some light on their structure and properties, but there's actually something deeper going on here.

The fact that the standard `Arrow`

class uses `(,)`

as monoidal structure is not coincidental: Hask is a cartesian closed category, so to embed Haskell's simply typed λ-calculus into some other category structure, we need at the very least a way to transport cartesian products, i.e. a premonoidal functor [1].

However, as long as our monoidal structure is comultiplicative and symmetric, we can always recover a first-order fragment of *λ*-calculus inside the "guest" category, and we don't even need an identity-on-objects functor [2].

The idea is that we can use the monoidal structure of the guest category to represent contexts, where weakening is given by `counit`

, contraction by `comult`

, and exchange by `swap`

.

There is an experimental GHC branch with a preprocessor which is able to translate expressions written in an arbitrary guest language into Haskell, given instances of appropriate type classes , which correspond exactly to the ones we have defined above.

## Examples

This exposition was pretty abstract, so we end with some examples.

We first need to define a few wrappers for our monoidal combinators, so we don't have to deal with the `PipeC`

newtype:

```
> split :: Monad m => Pipe a (Either a a) m r
> split = unPipeC comult
>
> join :: Monad m => Pipe (Either a a) a m r
> join = unPipeC mult
>
> (*+*) :: Monad m => Pipe a b m r -> Pipe a' b' m r
> -> Pipe (Either a a') (Either b b') m r
> f *+* g = unPipeC $ bimap (PipeC f) (PipeC g)
>
> discardL :: Monad m => Pipe (Either Void a) a m r
> discardL = unPipeC idl
>
> discardR :: Monad m => Pipe (Either a Void) a m r
> discardR = unPipeC idr
```

Now let's write a `tee`

combinator, similar to the tee command for shell pipes:

```
> tee :: Monad m => Pipe a Void m r -> Pipe a a m r
> tee p = split >+> firstP p >+> discardL
>
> printer :: Show a => Pipe a Void IO r
> printer = forever $ await >>= lift . print
>
> ex6 :: IO ()
> ex6 = do
> (sourceList [1..5] >+>
> tee printer >+>
> (fold (+) 0 >>= yield) $$
> printer)
> return ()
> {- ex6 == mapM_ print [1,2,3,4,5,15] -}
```

Another interesting exercise is reimplementing the `groupBy`

combinator of the previous post:

```
> groupBy :: Monad m => (a -> a -> Bool) -> Pipe a [a] m r
> groupBy p =
> -- split the stream in two
> split >+>
>
> -- yield Nothing whenever (not (p x y))
> -- for consecutive x y
> ((consec >+>
> filter (not . uncurry p) >+>
> pipe (const Nothing)) *+*
>
> -- at the same time, let everything pass through
> pipe Just) >+>
>
> -- now rejoin the two streams
> join >+>
>
> -- then accumulate results until a Nothing is hit
> forever (until isNothing >+>
> pipe fromJust >+>
> (consume >>= yield))
>
> -- yield consecutive pairs of values
> consec :: Monad m => Pipe a (a, a) m r
> consec = await >>= go
> where
> go x = await >>= \y -> yield (x, y) >> go y
>
> ex7 :: IO ()
> ex7 = do (sourceList [1,1,2,2,2,3,4,4]
> >+> groupBy (==)
> >+> pipe head
> $$ printer)
> return ()
> {- ex7 == mapM_ print [1,2,3,4] -}
```

## References

[1] J. Power and E. Robinson, “Premonoidal categories and notions of computation,” *Mathematical. Structures in Comp. Sci.*, vols. 7, pp. 453-468, 1997.

[2] A. Megacz, “Multi-Level Languages are Generalized Arrows,” *arXiv:1007.2885*, 2010.