# Diagonalization/ dupe for monads and tuples?

Wed Sep 16 22:35:56 UTC 2020

```Whoops, I just realized I've been responding to Carter specifically

I was having some trouble understanding the `unJoin` stuff but I read it
a few more times and I think I understand it a little better now.

In my personal experience the "abstracted version" of `x -> (x, x)` I
use most often looks like this:

```

class SymmetricMonoidal t i p => CartesianMonoidal t i p

where

duplicate :: p x (x `t` x)

-- Laws:

-- duplicate >>> first  discard = fwd lunit

-- duplicate >>> second discard = fwd runit

-- where

-- lunit :: Monoidal t i p => Iso p x (i `t` x)

-- runit :: Monoidal t i p => Iso p x (x `t` i)

```

structure gives us a cartesian monoidal structure.

This doesn't really seem to be relevant to what you folks are looking
for, but it does bring up a question. If some `Bifunctor` `t` happens to
form a monoidal structure on `->`, wouldn't the existence of appropriate
`forall a. a -> t a a` and `forall a. x -> Unit t` functions pigeonhole
it into being "the" cartesian monoidal structure on `->` (and thus
naturally isomorphic to `(,)`)?

On 9/16/20 5:26 PM, Emily Pillmore wrote:
> 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
>
> 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 <mailto: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 at gmail.com <mailto:strake888 at gmail.com>> wrote:
>
>         We also have
>
>
>
>         diag = join bipure
>
>
>
>
>
>
>         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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