[Haskell-cafe] When is a composition of catamorphisms a catamorphism?
wren ng thornton
wren at freegeek.org
Sun Aug 26 07:33:20 CEST 2012
On 8/24/12 3:44 AM, Sebastien Zany wrote:
> More specifically (assuming I understood the statement correctly):
> Suppose I have two base functors F1 and F2 and folds for each: fold1 :: (F1
> a -> a) -> (μF1 -> a) and fold2 :: (F2 a -> a) -> (μF2 -> a).
> Now suppose I have two algebras f :: F1 μF2 -> μF2 and g :: F2 A -> A.
> When is the composition (fold2 g) . (fold1 f) :: μF1 -> A a catamorphism?
From <http://comonad.com/haskell/catamorphisms.html> we have the law:
F, a functor
G, a functor
e, a natural transformation from F to G
(i.e., e :: forall a. F a -> G a)
g, a G-algebra
(i.e., f :: G X -> X, for some fixed X)
it follows that
cata g . cata (In . e) = cata (g . e)
The proof of which is easy. So it's sufficient to be a catamorphism if
your f = In . e for some e. I don't recall off-hand whether that's
necessary, though it seems likely
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