Are Haskell monads stronger than strong monads?

Question

Haskell's monads are usually considered to mean strong monads in category theory, but it seems like the former is a bit stronger than the latter. With strong monads, you have a Kleisli extension operator

$-^* : C(a, Tb) → C(Ta, Tb)$

that operates outside the category $C$, whereas with Haskell's monads you can write an internal extension operator

$e : \mathbf{Hask}(a ⇒ Tb, Ta ⇒ Tb)$

that operates entirely inside $\mathbf{Hask}$. The intuitive difference is that $M^*$ could involve a translation on $M : C(a, Tb)$ that inspects its syntax, whereas $eM$ can only operate on $M$ extensionally, only manipulating $M$ through whatever accessors are available for its type.

AFAICT, $-^*$ can be derived from $e$, but not vice versa, even assuming tensorial strength and Cartesian-closed structure. Is this true, or is there a way to derive $e$ from $-^*$ that I missed?

If Haskell's monads are not just strong monads, then do they have a name, and is there any interesting difference known about them vs. strong monads?

Answer 1

In a CCC, we have the following equivalence:

$$C(a ⇒ Tb, Ta ⇒ Tb) \cong C((a ⇒ Tb)×Ta, Tb)$$

Using tensorial strength, to give an element of the latter it suffices to give an element of:

$$C(T((a ⇒ Tb) × a), Tb)$$

And then using the evaluation map, it suffices to give:

$$C(TTb, Tb)$$

But this is given by the multiplication of the monad. So, strong monads in a CCC actually do have this sort of internal 'bind' operation. Despite the presentation, this is actually one of the core motivations behind strong monads/functors—internalizing the operations. The tensorial strength is just a characterization that makes sense without the category necessarily being closed.