I'm playing around with type-aligned sequences, and in particular I'm messing around with the idea of folding them. A foldable type-aligned sequence looks something like this:

```
class FoldableTA fm where
foldMapTA :: Category h =>
(forall b c . a b c -> h b c) ->
fm a b d -> h b d
foldrTA :: (forall b c d . a c d -> h b c -> h b d) ->
h p q -> fm a q r -> h p r
foldlTA :: ...
```

It's pretty easy to implement `foldrTA`

in terms of `foldMapTA`

by first using `foldMapTA`

to convert the sequence to a type-aligned list in the naive way (i.e., using the type-aligned list category) and then folding over that list. Unfortunately, this can be quite inefficient because long lists may be prepended to short ones. I've been trying to figure out a way to use a trick similar to the one used in `Data.Foldable`

to define the right and left folds more efficiently, but the types are making me dizzy. `Endo`

does not seem general enough to do the trick, and every step I take in other directions leads me to more type variables than I can keep track of.

Answer:

I found that this typechecks:

```
{-# LANGUAGE RankNTypes #-}
module FoldableTA where
import Control.Category
import Prelude hiding (id, (.))
class FoldableTA fm where
foldMapTA :: Category h => (forall b c . a b c -> h b c) -> fm a b d -> h b d
foldrTA :: (forall b c d . a c d -> h b c -> h b d) -> h p q -> fm a q r -> h p r
foldrTA f z t = appEndoTA (foldMapTA (\x -> TAEndo (f x)) t) z
newtype TAEndo h c d = TAEndo { appEndoTA :: forall b. h b c -> h b d }
instance Category (TAEndo h) where
id = TAEndo id
TAEndo f1 . TAEndo f2 = TAEndo (f1 . f2)
```

No idea if it makes any sense, but with so many type indexes around, I doubt that there is much type-checking code that does *not* make sense.

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