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Stream: learning: questions

Topic: tensor-hom duality for commutative monoids

John Baez (May 13 2025 at 09:56):

Does someone know a reference for this?

Given two commutative monoids $M,N$ the set of monoid homomorphisms $\text{hom}(M,N)$ becomes a commutative monoid with pointwise addition, and there's also a tensor product of commutative monoids with a natural isomorphism

$\text{hom}(M\otimes N,L)\cong\text{hom}(M,\text{hom}(N,L)).$

I am not primarily wanting a way to deduce this from more general facts, nor do I want to be told that it's easy to check (which is true). I just want to use it and refer to a proof.

If I needed it only for abelian groups I'd know plenty of textbooks to cite - though it's so well-known I probably wouldn't bother. But for some reason I don't know analogous books or papers for commutative monoids.

John Berman stated it without proof on MathOverflow, but that doesn't help me.

James Deikun (May 13 2025 at 11:14):

I haven't seen a reference for commutative monoids specifically, but it's a general fact about commutative/monoidal algebraic theories/monads ...

Nathanael Arkor (May 13 2025 at 11:42):

I think it would be appropriate to cite Kock's paper "Closed categories generated by commutative monads" together with a reference that the free commutative monoid monad is commutative, e.g. Meseguer--Montanari's "Petri nets are monoids".

John Baez (May 13 2025 at 16:32):

Okay, thanks. I was hoping someone had written a nice book or paper on commutative monoids that covers a lot of material we often see presented for abelian groups. (Or a book on modules of rigs that covers a lot of material we often see presented for modules of rings!) Someday such books will exist, but maybe not yet.

John Baez (May 13 2025 at 16:39):

So, I'll point readers to the more general results on commutative monads.

John Baez (May 13 2025 at 16:47):

I see Montanari and Meseguer write

It is well known [emphasis mine] that a tensor product $A \otimes B$ can be defined in the category $\mathsf{CMon}$ of commutative monoids so that, up to natural isomorphisms, $\otimes$ is associative, commutative, and has as an identity. It is also well known that the monoid homomorphisms from A to B form a commutative monoid [A, B] and that there is a natural isomorphism

$\mathsf{CMon}(A \otimes B, C) \cong \mathsf{CMon}(A, [B ,C])$

in other words, the category of commutative monoids is a symmetric
monoidal closed category (MacLane, 1971). This is just like a Cartesian
closed category except that....

Nathanael Arkor (May 13 2025 at 17:23):

Authors using "well known" without providing a reference is a pet peeve of mine :slight_smile:

Madeleine Birchfield (May 13 2025 at 19:32):

It is well known that a monad is a monoid in the category of endofunctors.

John Baez (May 13 2025 at 20:23):

Nathanael Arkor said:

Authors using "well known" without providing a reference is a pet peeve of mine :slight_smile:

Even worse is what I'll be doing by citing Meseguer and Montanari: saying that something is well known, and providing a reference.... to someone else who says it's well known! But Kock's paper provides proofs, I believe (at a higher level of generality).

Nathanael Arkor (May 14 2025 at 06:54):

Well, Meseguer and Montanari at least explain why the commutative monoid monad is commutative, from which the facts you want follows from Kock's paper. So citing both references ought to cover all the bases.

Mike Shulman (May 14 2025 at 07:06):

The nLab page [[tensor product of commutative monoids]] at least contains a construction of the tensor product, although not the internal-hom and their adjunction. Maybe someone can add that, and then it can be a reference. (-:

John Baez (May 14 2025 at 08:53):

I'm planning to cite M&M and Kock, and say as little as possible about the details. I'm thinking of running an undergraduate project where the students prove basic theorems like this about commutative monoids and write a nice little paper about them. (Teaching students to write would be the tiresome part.)

P. A. Grillet's book Commutative Semigroups mentions a couple papers about tensor products of commutative semigroups, but doesn't develop them.

(I'm not sure exactly what happens when you write a large book of theorems about commutative semigroups instead of commutative monoids.)

Todd Trimble (May 20 2025 at 12:45):

John Baez said:

Does someone know a reference for this?

Given two commutative monoids $M,N$ the set of monoid homomorphisms $\text{hom}(M,N)$ becomes a commutative monoid with pointwise addition, and there's also a tensor product of commutative monoids with a natural isomorphism

$\text{hom}(M\otimes N,L)\cong\text{hom}(M,\text{hom}(N,L)).$

I am not primarily wanting a way to deduce this from more general facts, nor do I want to be told that it's easy to check (which is true). I just want to use it and refer to a proof.

If I needed it only for abelian groups I'd know plenty of textbooks to cite - though it's so well-known I probably wouldn't bother. But for some reason I don't know analogous books or papers for commutative monoids.

John Berman stated it without proof on MathOverflow, but that doesn't help me.

This may be an unpopular opinion, but this is a case where I would not blush to say "it is well-known that". Perhaps with a parenthetical "the proof of this is just like the proof for abelian groups".

But in response to Mike's suggestion, which gives another option, I've now added the stuff about the internal homs and the adjunction to [[tensor product of commutative monoids]]. To me, this is the whole point of tensor products in the first place: internal homs $[B, C]$ are defined pointwise by commutativity in $C$, and then the endofunctors $[B, -]$ are closed under composition, up to isomorphism, by $[A, -] \circ [B, -] \cong [A \otimes B, -]$.

John Baez (May 20 2025 at 13:03):

Thanks, Todd! I'm glad you added it to the nLab, because it really deserves to be there. I'm a bit reluctant to cite the nLab since it's time-dependent. Maybe I shouldn't be.

What I actually did was add the following to the paper @Adittya Chaudhuri and I are writing.

This [the theorem we're trying to prove] can be shown directly, but we sketch a proof using some facts, well known for abelian groups, which have easy analogues for commutative monoids [MeseguerMontanari1990] and even more general algebraic structures [Kock1971]. First, given $A, B \in \mathsf{CommMon}$, the set of homomorphisms $A \to B$ can be made into a commutative monoid $[A,B]$ using pointwise operations:

$(f + g)(a) = f(a) + g(a) , \qquad \forall f,g \in [A,B], \; a \in A .$

Second, we can define a tensor product $A \otimes B$ of commutative monoids such that homomorphisms $A \otimes B \to C$ correspond naturally to maps $A \times B \to C$ that are homomorphisms in each argument. This tensor product obeys hom-tensor adjointness: that is, there is a natural isomorphism

$\mathsf{CommMon}(A \otimes B, C) \cong \mathsf{CommMon}(B, [A,C]) .$

John Baez (May 20 2025 at 13:04):

This is a bit long, but I figure it's good to spell things out for anyone who doesn't already know this stuff, with enough detail that they can fill in the holes themselves.

Nathanael Arkor (May 20 2025 at 13:38):

John Baez said:

Thanks, Todd! I'm glad you added it to the nLab, because it really deserves to be there. I'm a bit reluctant to cite the nLab since it's time-dependent. Maybe I shouldn't be.

You can cite specific revisions of the nLab, e.g. https://ncatlab.org/nlab/revision/tensor+product+of+commutative+monoids/8.