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

[Olaf Klinke]

On Sat, 2021-01-09 at 23:45 +0100, MigMit wrote:
But it won't be cartesian closed. If it were, then for any finite
> > > X
> > > and Y we should have
> > > 
> > > Maybe (X^Y) ~
> > > () -> Maybe (X^Y) ~
> > > OneOrBoth () Y -> Maybe X ~
> > > (() -> Maybe X, Y -> Maybe X, ((), Y) -> Maybe X) ~
> > > (Maybe X, Y -> Maybe X, Y -> Maybe X)
> > > 
> > > so
> > > 
> > > X^Y ~ (X, Y -> Maybe X, Y -> Maybe X)
> > > 
> No, my arrows and isomorphisms are all in the original category, not
> the Kleisli one — although the "X^Y" is the exponent in the Kleisli
> category.

I don't follow your argument. I still must be misinterpreting
something. 

  Maybe (X^Y)
~ () -> Maybe (X^Y)          
               -- because () is terminal in Hask
~ OneOrBoth () Y -> Maybe X  
               -- OneOrBoth is the product in Kleisli Maybe
~ (Maybe X, Y -> Maybe X, Y -> Maybe X)
               -- universal property of coproduct

But how did you get to the next step: 

X^Y ~ (X, Y -> Maybe X, Y -> Maybe X)

I think this can not hold for cardinality reasons: 

1+X*((1+X)^Y)^2 /= (1+X)*((1+X)^Y)^2

Olaf