# [Haskell-cafe] A quick question about distribution of finite maps.

Tom Smeding x at tomsmeding.com
Sat Jan 9 09:21:29 UTC 2021

```Hi Ignat,

A ghci session:

Prelude> import Test.QuickCheck
Prelude Test.QuickCheck> import qualified Data.Map.Strict as Map
Prelude Test.QuickCheck Map> import Data.Map.Strict (Map)
Prelude Test.QuickCheck Map Data.Map.Strict> let distribute = Map.unionsWith Map.union . Map.mapWithKey (fmap . Map.singleton)
Prelude Test.QuickCheck Map Data.Map.Strict> quickCheck \$ \m -> distribute (distribute m) == (m :: Map Int (Map Int Int))
*** Failed! Falsified (after 2 tests and 2 shrinks):
fromList [(0,fromList [])]
Prelude Test.QuickCheck Map Data.Map.Strict> distribute (Map.singleton 1 mempty)
fromList []

It seems your property fails with the map `fromList [(0, fromList [])]`. I'm not sure if you need it to work for that case; I'm not a category theorist.

Cheers,
Tom

‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐
On Saturday, January 9, 2021 9:38 AM, Ignat Insarov <kindaro at gmail.com> wrote:

> Hello!
>
> Finite maps from `"containers" Data.Map` look like they may form a
> Cartesian closed category. So it makes sense to ask if the rule α ⇸
> (β ⇸ γ) ≡ ⟨α; β⟩ ⇸ γ ≡ ⟨β; α⟩ ⇸ γ ≡ β ⇸ (α ⇸ γ) that holds in suchcategories does hold for finite maps. Note that, a map being a
> functor, this also may be seen as f (g α) ≡ g (f α), which would
> work if maps were `Distributive` [1].
>
> It looks to me as though the following definition might work:
>
> distribute = unionsWith union . mapWithKey (fmap . singleton)
>
> — And it works on simple examples. (I checked the law `distribute ∘ distribute ≡ id` — it appears to be the only law required.)
>
> Is this definition correct? Is it already known and defined elsewhere?
>
> [1]: https://hackage.haskell.org/package/distributive-0.6.2.1/docs/Data-Distributive.html#t:Distributive
>
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```

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