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There is a free commutative monoid monad. There should also be a free monoid monad. There should be maps between the monads. How do we define maps between monads? There should be a forgetful functor between the category of free monoids and the category of free commutative monoids. What is the relationship between the functor and the map between the monads?

I think you can take a free commutative monoid (say finitely presented) and just forget the axiom at $ab=ba$ and that should give you the functor from free commutative monoids to free monoids. Does that sound right? I imagine that you can have a category of monads, and a map between them should be a natural transformation between the endofunctors of the monads, but you also need higher order maps between all of the natural transformations. That data should specify the map. It sounds like somethign that should live in an n-category where n is suitably chosen. I could use some help though. And then further, there should be some kind of equivalence between the maps in the category of monads and the functors between the categories, but I don't know what the theory of that is.

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    $\begingroup$ Do you have a guess of what a morphism between monads is? You say that there "should" be a functor from the free monoids to the category of free commutative monoids: you do not have any idea of what that functor is? $\endgroup$ – Mariano Suárez-Álvarez Jan 26 '18 at 18:53
  • $\begingroup$ I think you can take a free commutative monoid (say finitely presented) and just forget the axiom at $ab=ba$ and that should give you the functor from free commutative monoids to free monoids. Does that sound right? I imagine that you can have a category of monads, and a map between them should be a natural transformation between the endofunctors of the monads, but you also need higher order maps between all of the natural transformations. That data should specify the map. It sounds like somethign that should live in an n-category where n is suitably chosen. I could use some help though. $\endgroup$ – Ben Sprott Jan 26 '18 at 19:58
  • $\begingroup$ My approach would be to go to the prototypical example of monad, and ask: given $M \to N$ a morphism of monoids, what can you get in terms of the monads $M \times \cdot$ and $N \times \cdot$? So, there would be a morphism of functors $M \times \cdot \to N \times \cdot$ - what properties of this morphism would correspond to (or be equivalent to) $M \to N$ being a morphism of monoids? $\endgroup$ – Daniel Schepler Jan 26 '18 at 20:05
  • $\begingroup$ And then, given that this would induce a functor from $N$-sets to $M$-sets, can you give a similar description based on such a functor of the functor between the Eilenberg-Moore categories of monad algebras? $\endgroup$ – Daniel Schepler Jan 26 '18 at 20:07
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    $\begingroup$ Wow. Bringing in n-categories already?! I suggest you think what a morphism of monads is before going the hard drug route... $\endgroup$ – Mariano Suárez-Álvarez Jan 27 '18 at 21:45

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