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An important fact about abelian monoids is that, for any two abelian monoids M and N, the homomorphisms from M to N themselves comprise an abelian monoid Hom(M, N), with monoid operation given by pointwise application of the monoid operation in N. (More generally, M here needn’t be an abelian monoid, but could be any monoid, or even any category at all, but nevermind that for now; this simply amounts to taking the free abelian monoid with a map from M and then proceeding as before). Thus, the category of abelian monoids is closed. (The key thing is just that combining two monoid homomorphisms via pointwise product yields another monoid homomorphism, when everything is abelian)
It seems to me this should generalize just as well to symmetric monoidal categories: for any two symmetric monoidal categories M and N, the symmetric monoidal functors from M to N comprise a functor category Hom(M, N), which it seems to me can be equipped with the structure of a symmetric monoidal category by taking the monoidal product to be pointwise application of the monoidal product in N. Thus, the category of symmetric monoidal categories is closed. (The key thing is just that combining two symmetric monoidal functors via pointwise product yields another symmetric monoidal functor)
And so on, this should generalize up the dimensionality ladder, and perhaps generalize in other directions as well. For instance, perhaps I was too conservative in considering only symmetric monoidal categories, and some version of this applies to braided monoidal categories as well? But I have not yet understood this well enough to understand its appropriate scope.
Anyway, I was hoping to find more information on this phenomenon somewhere on nLab, but did not know where to look. Could someone point me to any relevant articles or parts of articles discussing this phenomenon, if they exist?
Perhaps the notion of a Cartesian monad (or more generally a monoidal monad) is what you are looking for? I.e. a monad with sufficient properties/structure to ensure that the category of algebras for it is Cartesian closed (more generally closed monoidal)?
I think some details could be added to the nLab about this, though it is touched on here and there.
This monadic point of view should categorify to cover symmetric monoidal categories, but I don’t know if it has been worked out.
You may be looking for Pseudo-commutative monads and pseudo-closed 2-categories by Hyland and Power.
(I don’t think there is a “cartesian” version of the notion of monoidal/commutative monad, though. I’ve never heard of one.)
Thanks for the references!
I guess one thing I’d want to see confirmed is that symmetric monoidal n-categories have this property for all n, in some uniform way. And then the next thing I’d be interested in is to what extent one can weaken (or perhaps cannot at all weaken) from requiring full symmetry here (e.g., does any version of this work out for braided monoidal categories, or no dice?).
I am unfortunately still rather weak at thinking about these higher-dimensional phenomena myself, but perhaps with the light of someone else having already worked it out, I could follow it…
Re #4: I was thinking of this paper of Johnstone for an example of the Cartesian terminology.
Re #5: I would imagine that if how to work uniformly with coherences for all was understood, then one would have a uniform way to approach this monadic theory too. There should be a version for the braided, sylleptic, etc, cases as well, but I don’t know if the 2-category of braided monoidal categories itself will fit exactly into the pattern.
Thanks for that link. Following a citation it looks like this paper is even more relevant: the category of algebras for a product-preserving monad on a cartesian closed category is again cartesian closed. I didn’t know that! We should add it to some page on the lab. Unfortunately our pages monoidal monad, strong monad, and commutative algebraic theory are (in my opinion) a bit of a mess.
If you don’t get any answers here about symmetric monoidal -categories, you could try mathoverflow; there are a good deal of folks there who know the state of the art on stuff like that (probably you’re more likely to get answers about symmetric monoidal -categories, but the -categories live inside those).
I doubt that it works for the braided case, but I haven’t thought about it.
I agree that those pages are a bit of a mess. I will have a go at adding some things if no-one gets there before me :-).
In particular I don’t think that commutative monad ought to redirect to commutative algebraic theory; it ought to go to a page that’s directly about monads (on general categories, not just ). I’m kind of inclined to suggest that it be a page of its own, but I can see an argument for redirecting it to monoidal monad instead as long as that page is very upfront about the equivalence between the two notions.
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