# [Haskell-cafe] When is a composition of catamorphisms a catamorphism?

Sebastien Zany sebastien at chaoticresearch.com
Mon Aug 27 03:10:05 CEST 2012

```Thanks Wren. That was my guess too, but it seems not necessary:
http://stackoverflow.com/questions/12103309/when-is-a-composition-of-catamorphisms-a-catamorphism

On Sat, Aug 25, 2012 at 10:33 PM, wren ng thornton <wren at freegeek.org>wrote:

> 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?
>>
>
> we have the law:
>
>     Given
>         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
>
> --
> Live well,
> ~wren
>
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