Diagonalization/ dupe for monads and tuples?

[Emily Pillmore]

Just to clarify, that's not the "real" diagonal in the space, but it is a super useful translation that I'd like for free :)

On Wed, Sep 16, 2020 at 9:08 PM, Emily Pillmore < emilypi at cohomolo.gy > wrote:

> 
> @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 (
> masaeedu at 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 ( 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
>>>>> 
>>>>> 
>>>>> 
>>>>> 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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>>>> 
>>> 
>>> 
>>> 
>>> 
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>> 
>> 
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