# Diagonalization/ dupe for monads and tuples?

Emily Pillmore emilypi at cohomolo.gy
Wed Sep 16 21:26:56 UTC 2020

```Nice!

That's kind of what I was going for with Carter earlier in the day, thanks Matthew.

I think a diagonalization function and functor are both very sensible additions to `bifunctors` and `Data.Bifunctor`. The theory behind this is sound: The diagonalization functor Δ: Hask → Hask^Hask, forms the center of the adjoint triple `colim -| Δ -| lim : Hask → Hask^Hask`.

Certainly the function `diag :: a → (a,a)` is something I've seen written in several libraries, and should be included in `Data.Tuple` as a `base` function. The clear generalization of this function is `diag :: Biapplicative f ⇒ a → f a a`. I'm in favor of both existing in their separate capacities.

Thoughts?

Emily

On Wed, Sep 16, 2020 at 3:49 PM, Carter Schonwald < carter.schonwald at gmail.com > wrote:

>
> Is the join bipure definition taking advantage of the (a->) monad
> instance?  Slick!
>
>
>
>
> On Wed, Sep 16, 2020 at 3:39 PM Matthew Farkas-Dyck < strake888@ gmail. com
> ( strake888 at gmail.com ) > wrote:
>
>
>> We also have
>>
>>
>>
>> diag = join bipure
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>>
>>
>>
>>
>>
>> diag = unJoin . pure
>>
>> where
>>
>> newtype Join f a = Join { unJoin :: f a a } deriving (Functor)
>>
>> deriving instance Biapplicative f => Applicative (Join f)
>>
>>
>>
>> The latter seems on its face potentially related to the instance for
>>
>> lists of fixed length, but i am not sure how deep the connection may
>>
>> be.
>>
>>
>
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