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Stream: learning: questions

Topic: Two monads on the same functor

Ralph Sarkis (Mar 13 2024 at 17:58):

Is there a nice example of a functor with two monad structures on it (preferably with the same unit. EDIT: I think that's dumb because the unit of an adjunction determines the counit and hence the multiplication of the monad. EDIT2: now I think the first edit is dumb thanks to Peva's answer below and because a monad can arise from two different adjunctions)?

Ralph Sarkis (Mar 13 2024 at 17:58):

Related question: Any monoid in Set has an opposite monoid. I guess this is true in any symmetric (maybe braided is enough) monoidal category, but is there an opposite construction for monoids in other monoidal categories, possibly yielding a notion of opposite monad?

Peva Blanchard (Mar 13 2024 at 18:07):

Let $F : Set \rightarrow Set$ defined by $F A = \mathbb{R}_+ \times A$. Because the set $\mathbb{R}_+$ can be endowed with different monoid structures (e.g., usual multiplication, or addition, or maximum, etc.), you get different monad structures on $F$ accordingly.

But it's true that they don't have the same units.

Peva Blanchard (Mar 13 2024 at 18:13):

oh actually, max and addition have both 0 as neutral element.

John Baez (Mar 13 2024 at 18:52):

Peva Blanchard said:

Let $F : Set \rightarrow Set$ defined by $F A = \mathbb{R}_+ \times A$. Because the set $\mathbb{R}_+$ can be endowed with different monoid structures (e.g., usual multiplication, or addition, or maximum, etc.), you get different monad structures on $F$ accordingly.

That"s nice. And to highlight what's going on here, we could choose any set S and let FA = S×A. Nonisomorphic =monoid structures on S should give nonisomorphic monads, so already when S has 2 elements we get two, and when S is infinite I bet we get uncountably many.

(Are there uncountably many isomorphism classes of countable monoids? I don't know, but I bet there are.)

Matteo Capucci (he/him) (Mar 14 2024 at 08:50):

Another example should be powerset with intersection and top element vs powerset with union and bottom element. Of course these are isomorphic but definitely not the same!

Naïm Favier (Mar 14 2024 at 09:29):

for a more programmer-brained example, there's a whole zoo of monad structures on the free monoid (list) endofunctor: see https://hackage.haskell.org/package/exotic-list-monads-1.1.1/docs/Control-Monad-List-Exotic.html