This question on math.stackexchange piqued my interest. It asks what the monads on a given monoid are. That is, if we treat a monoid as a category with one object we can ask what the monads on that category are. I would have thought that this question would have had an elegant answer, because monads and monoids are both so fundamental. But in fact I can’t find a nice characterisation.
Writing down the definitions we find that a monad on a monoid is equivalent to “an endomorphism together with two elements such that:
, ,
, ,
,
.”
Now, in the nice cases when is a group or commutative, one can prove that and that is just the inner automorphism given by conjugation by . But in the general case I’m not able to prove that can still only be an inner automorphism. So does anyone know what kind of structure this is?
]]>In the discussion at the bottom of monoidal category, we read:
In fact a strict monoidal category is just a monoid internal to the category Cat. Unfortunately this definition is circular, since to define a monoid internal to Cat, we need to use the fact that Cat is a monoidal category!
And then later
For example, you can define a monoidal category to be a pseudomonoid internal to the 2-category Cat — but nobody knew how to define these concepts until they knew what a monoidal category is!
Doesn’t the same circularity afflict the definition of monoidal category that’s on the page? For example, the associator is given as
But this doesn’t make sense if taken literally. You cannot have a natural transformation between functors on different domains, and the domain of these functors are not the same. The domain of the left functor is whereas the domain of the right functor is . Of course those two categories are isomorphic, and using that isomorphism, we can make sense of the definition. But that’s using the monoidal structure of ! We’re being circular in exactly the same way as we would if we defined a monoidal category as a (pseudo)monoid in the monoidal (2-)category !
I guess it’s not circular in any formal sense, since we can just observe that any cartesian category has canonical isomorphisms and we can just insert that into the definition as needed, without commenting that it is part of a monoidal structure on the ambient category. The same applies to any monoid in any cartesian category. In particular, I think it’s not formally circular to define a (weak/strict) monoidal category as a (pseudo) monoid in .
Shouldn’t that equivalent definition be mentioned higher in the article, since it’s valid and not really circular?
And shouldn’t the article be more explicit about this, about using the cartesian associator and unitors of , given that it’s basically an article about the need to be careful and rigorous about the axioms of associators and monoidal structures?
Also, is there some more coherent, higher categorical way out of this circularity, other than than just capping it with a cartesian structure at some level of the higher categorical ladder?
]]>On the nLab there are several links to a (nonexistent) page monoid object which are then redirected to monoid, while I think they should be redirected to monoid in a monoidal category, since we are talking about internal monoids. How can this be done systematically?
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