@Asad that's a perfectly reasonable way to think about diagonal operations: as the data of a cartesian monoidal category, and the laws are correct in this case. I think we can get away with the additional abstraction to `Biapplicative` in this case, though.

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 `(,)`)?


Only if you chose that particular unit and that particular arrow. But there are other places where having a general `Biapplicative` contraint would make it useful. For example,  i'd like to use this in `smash` with `diag :: a → Smash a a`, defining the adjoining of a point to `a` and creating a diagonal in the subcategory of pointed spaces in Hask, so that I don't have to redefine this as `diag' = quotSmash . view _CanIso . diag . Just`.

Cheers,
Emily







On Wed, Sep 16, 2020 at 6:35 PM, Asad Saeeduddin <masaeedu@gmail.com> wrote:

Whoops, I just realized I've been responding to Carter specifically instead of to the list.

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)
  discard :: p x i


-- 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)

```

i.e. adding a suitable duplicate and discard to some symmetric monoidal 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 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@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> wrote:
We also have



diag = join bipure



and (in pseudo-Haskell)



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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