## February 14, 2024

### Cartesian versus Symmetric Monoidal

#### Posted by John Baez

James Dolan and Chris Grossack and I had a fun conversation on Monday. We came up some ideas loosely connected to things Chris and Todd Trimble have been working on… but also connected to the difference between classical and quantum information.

I’ve long been fascinated by the relation between ‘classical’ and ‘quantum’. One way this manifests is the relation between cartesian monoidal categories (like the category of sets with its cartesian product) and more general symmetric monoidal categories (like the category of Hilbert spaces with its tensor product).

Cartesian monoidal categories let us ‘duplicate and delete data’ since every object $x$ comes with morphisms

$\Delta_x : x \to x \otimes x \;\; and \;\; \epsilon_x: x \to I$

where $I$ is the unit object. These obey equations making $x$ into a cocommutative comonoid — just like a commutative monoid, only backwards. For example if you duplicate some data, it should make no difference if you then switch the two copies. Moreover, in a cartesian monoidal category $\Delta$ and $\epsilon$ are natural transformations. In quantum mechanics, duplication and deletion of data in a natural way is generally impossible.

Given this, it’s interesting that we can force any symmetric monoidal category to become cartesian. I believe can do it in two ways, which are left and right adjoint to the forgetful map sending cartesian monoidal categories to their underlying symmetric monoidal categories. Moreover, I conjecture that we can describe both these ways very neatly using the free cartesian monoidal category on one object, which I call $F$. If these conjectures are right, this category has the power to make any symmetric monoidal category become cartesian!

#### The details, such as they are

We’ve got two 2-categories:

• the 2-category of symmetric monoidal categories, $SMC$.
• the 2-category of cartesian monoidal categories, $Cart$.

There’s an obvious forgetful 2-functor $U: Cart \to SMC$. I believe this has both left and right adjoints, in a suitable 2-categorical sense. These are called ‘pseudoadjoints’. So:

Conjecture 0. The forgetful 2-functor $U: Cart \to SMC$ has a left pseudoadjoint $L: SMC \to Cart$ and a right pseudoadjoint $R: SMC \to Cart$.

I claim that $R$ sends any symmetric monoidal category $C$ to the category of cocommutative comonoid objects in $C$: this category is cartesian by

Fox’s Theorem. A symmetric monoidal category is cartesian if and only if it is isomorphic to its own category of cocommutative comonoids. Thus every object is equipped with a unique cocommutative comonoid structure $\Delta_x : x \to x \otimes x$ and $\epsilon_x : x \to I$, and these structures are respected by all maps.

$L$ on the other hand should ‘freely’ make any symmetric monoidal category $C$ into a cartesian one. To do this, it should freely give each object $x$ morphisms $\Delta_x : x \to x \otimes x$ and $\epsilon_x : x \to I$ making it into a cocommutative comonoid, imposing equations to make sure every morphism is a comonoid homomorphism. That’s a bit vague, of course. So I want to describe an attempt to make this more precise.

Categorifying the usual tensor product of commutative monoids, there’s a tensor product $\boxtimes$ of symmetric monoidal categories. This has the universal property that if $C, D$ and $E$ are symmetric monoidal categories, functors

$f : C \times D \to E$

that are symmetric monoidal in each argument separately correspond to symmetric monoidal functors

$f: C \boxtimes D \to E$

The existence of this tensor product is a special case of a result of Hyland and Power. In fact their work shows this tensor product makes $SMC$ into a monoidal 2-category. I’m sure it must be symmetric monoidal in a suitable 2-categorical sense—but has anyone written that up?

Now for the fun part.

Conjecture 1. $L C \simeq F \boxtimes C$ where $F$ is the free cartesian category on one object, i.e. the initial Lawvere theory.

To construct $F$ we start with the category of finite sets with coproduct as its monoidal structure, which is the free cocartesian monoidal category on one object, and then take its opposite:

$F \simeq (FinSet, +)^{op}$

I’ve believed Conjecture 1 since at least 2006 (see page 59 here, where I rashly called it a ‘theorem’, probably because I worked out enough details to make it seem obvious). But now Chris, James and I guessed that the right pseudoadjoint $R: SMC \to Cart$ has a similar beautiful description!

Hyland and Power didn’t merely show the $\boxtimes$ product of symmetric monoidal categories makes $SMC$ into a monoidal 2-category. They also showed that this monoidal 2-category is closed in a suitable 2-categorical sense—or as they put it, ‘pseudo-closed’.

In other words, given symmetric monoidal categories $C, D$ and $E$, there is a symmetric monoidal category $[D,E]$ such that symmetric monoidal functors

$f: C \boxtimes D \to E$

correspond to symmetric monoidal functors

$f: C \to [D, E]$

If I understand this correctly, $[D,E]$ has symmetric monoidal functors $g: D \to E$ as objects, and symmetric monoidal natural transformations between these as morphisms. The tensor product on $[D,E]$ is defined ‘pointwise’.

And I claim:

Conjecture 2. $R C \simeq [F, C]$.

The idea is this: $F$ is not only the free cartesian monoidal category on one object. It’s also the free symmetric monoidal category on a cocommutative comonoid! Objects of $[F,C]$ are symmetric monoidal functors $f: F \to C$, so these should be the same as cocommutative comonoids in $C$. So, $[F,C]$ is the category of cocommutative comonoids in $C$, which is $R C$.

Has someone already proved these two conjectures? If not, I hope someone does.

Posted at February 14, 2024 6:46 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3515

### Re: Cartesian versus Symmetric Monoidal

Regarding whether or not anyone has written up the (appropriately 2-categorical) symmetric monoidal structure of SMC, I suspect you’re looking for Vincent Schmitt’s work: https://arxiv.org/abs/0711.0324.

Posted by: John Baez on February 14, 2024 7:48 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Todd Trimble pointed out on Zulip that Schmitt’s tensor product is on the 2-category of symmetric monoidal categories and lax symmetric monoidal functors, whereas John wants strong ones.

Another more recent followup to Hyland and Power’s work, which does use the strong functors, is John Bourke’s Skew structures in 2-category theory and homotopy theory, which uses skew-monoidal structures and model categories to manage the bureaucracy in the construction much more cleanly. Skimming Bourke’s paper I’m not entirely sure whether his results imply immediately that $SMC$ is a symmetric monoidal bicategory or not; he does have some results about things being “homotopy symmetric monoidal” but doesn’t appear to state a conclusion about “symmetric monoidal bicategories”.

Posted by: Mike Shulman on February 14, 2024 8:59 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Thanks! I definitely intend $SMC$ to be the 2-category of symmetric monoidal categories, strong symmetric monoidal functors and (symmetric) monoidal natural transformations. Similarly $Cart$ is the 2-category of cartesian monoidal categories, strong symmetric monoidal functors and (symmetric monoidal) natural transformations. I was waiting for someone to ask me if I meant strong or lax!

Posted by: John Baez on February 14, 2024 10:59 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

It sounds like you’re suggesting that $Cart$ should be the 2-category of $F$-modules, where $F$ is regarded as a (pseudo)monoid in the monoidal 2-category $SMC$. This would then imply your two conjectures by general facts about extension and coextension of scalars for modules in a closed monoidal (2-)category. This is a cute idea and I don’t think I’ve seen it suggested before.

An equivalent way to say that would be that a symmetric monoidal category $C$ is cartesian if and only if it comes with a symmetric monoidal functor $F \to [C,C]$ that is a morphism of (pseudo)monoids in $SMC$. It seems plausible to me that if we compile that out using a generators-and-relations presentation of $F$ in $SMC$ it would reduce to the characterization of cartesian monoidal categories in terms of natural duplication and deletion operations.

Posted by: Mike Shulman on February 14, 2024 8:45 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Yes, James did point out that all this stuff looks awfully reminiscent of things we usually first see in school when we learn about modules of rings: extension and coextension of scalars are left and right adjoint to restriction of scalars! And let me spell out the mental steps for everyone reading:

We don’t really need rings to set up the theory of modules, restriction of scalars, etc.: it’s enough to use monoids, and we can then categorify and use pseudomonoids. We can define modules of a pseudomonoid in a monoidal 2-category, and restriction of scalars makes sense. Then, if our 2-category is nice enough, the usual formulas for extension and coextension of scalars will also hold. The most-discussed example of a module of a pseudomonoid is an actegory of a monoidal category. A monoidal category is a pseudomonoid in $Cat$. But for what we’re doing now, we need the fact that $F$ is a pseudomonoid in $SMC$.

So it’s good to go the extra mile and say what you just said, or something like this:

Conjecture 3. The 2-category $Cart$ is equivalent to the 2-category of $F$-modules in $SMC$.

Posted by: John Baez on February 14, 2024 10:44 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

In me and Bruno’s work on actegories, we prove actions of $M$ on $C$ in SMC are given by symmetric monoidal functors $f:M \to C$.

So your conjecture becomes: ‘being cartesian’ for a symmetric monoidal category $M$ is equivalent to the data of a (then necessarily unique up to iso) symmetric monoidal functor $k:F \to M$.

Such a $k$ has a very constrained form, since it is basically determined by the data of $k(1)$. Then $k(n) \cong k(1)^{\otimes n}$. But this doesn’t seem to be so constrained as to enforce every two $k$ to be isomorphic.

So there’s either something to tweak in the definition or the theorem in my aforementioned work is wrong!

Posted by: Matteo Capucci on February 17, 2024 5:40 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

The dilemma is readily resolved: the result I cite works for action with respect to the cartesian product of symmetric monoidal categories, but here we are talking about their tensor product.

Posted by: Matteo Capucci on February 17, 2024 6:51 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Anything useful from this MO question?

Posted by: David Corfield on February 16, 2024 7:00 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Thanks! Yes, it mentions this paper:

This paper shows some version of what Mike was conjecturing: namely, that cartesian categories are the same as $F$-modules in the 2-category of symmetric monoidal categories. And there’s a lot of nice other stuff in it.

Posted by: John Baez on February 16, 2024 9:17 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Not the first time that paper has appeared at the $n$Café. Mike was pleased to see it back here.

I see amongst other work, Berman addresses other questions of yours:

Baez asks whether the Euler characteristic (defined for spaces with finite homology) can be reconciled with the homotopy cardinality (defined for spaces with finite homotopy). …This provides a negative answer to Baez’s question globally, but a positive answer when we restrict attention to a prime.

Posted by: David Corfield on February 17, 2024 8:45 AM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Alas, Berman’s paper on homotopy cardinality seems to have a significant mistake in it. In this paper:

Yanofski tries a different approach to unifying Euler characteristic and homotopy cardinality, and he writes in a footnote:

A solution using a different technique was claimed in the preprint [Ber18], but apparently it has a mistake. The proposed approach is however still interesting and merits further investigation.

Apparently Berman agrees that it’s a mistake and has removed the paper from his homepage and not tried to publish it. It would be nice if Berman added an erratum to the arXiv paper — it would prevent people from being misled. But I’m very happy that Berman and Yanofski have been trying to carry out this unification!

Here’s the abstract of Yanofski’s paper, by the way:

We answer a question of John Baez, on the relationship between the classical Euler characteristic and the Baez–Dolan homotopy cardinality, by constructing a unique additive common generalization after restriction to an odd prime $p$. This is achieved by $\ell$-adically extrapolating to height $n = -1$ the sequence of Euler characteristics associated with the Morava $K(n)$ cohomology theories for (any) $\ell | p-1$. We compute this sequence explicitly in several cases and incorporate in the theory some folklore heuristic comparisons between the Euler characteristic and the homotopy cardinality involving summation of divergent series.

Posted by: John Baez on February 17, 2024 11:37 PM | Permalink | Reply to this

### Re: Cartesian versus Symmetric Monoidal

Some remarks, rather brief for lack of time, which I hope will not appear too cryptic:

1. I think one of the great mistakes of category theory has been to encourage use of weak structures. If one rephrases the question to work entirely with strict structures (strict cartesian, strict symmetric monoidal), one loses none of its essence. For the remainder of my comment, I will assume such a strict rephrasing.

2. The question is intimately related to the relationship between various types of cubical sets, though the two types of cubical sets which correspond directly to the question (with diagonals, with symmetries) are not the most commonly studied.

3. One look at the question itself more cubically. Work with strict cubical $\infty$-categories (here strict cubical 2- or 3-categories are enough). Any flavour of cube will give rise to such a notion of strict cubical $\infty$-category, and all the varieties have a natural tensor product, which is basically the same (just more immediate) as the Gray tensor product in a globular setting. Symmetric monoidalness of a category can be expressed in terms of a functor $C \otimes C \rightarrow C$, where $\otimes$ is this tensor product, where one assumes that one’s cubes have at least symmetries (one requires that the functor preserves symmetries). Cartesian monoidalness can be expressed in exactly the same way, assuming this time that one’s cubes have at least diagonals.

4. Assume that one’s cubes have diagonals. It is more or less trivial to see that for any monoidal category, one can obtain a cartesian one by just chucking away any objects for which the defining functor does not preserve diagonals, and one obtains in this way a right adjoint to the inclusion of cartesian monoidal categories in monoidal ones. The proof is basically identical to the one proving that taking the core of a category is right adjoint to the inclusion of groupoids, and can in fact be understood in exactly the same way as I am describing here if one works just with intervals instead of cubes (there is no need for tensoring then). Anyhow, this right adjoint restricts to symmetric monoidal categories if one’s cubes have both diagonals and symmetries.

5. It is instructive to consider what the free strict cartesian category looks like, thinking in this same cubical way, and in terms of a functor $C \otimes C \rightarrow C$.

6. It is obvious, still thinking in this cubical way, that one can freely make any category cartesian monoidal by just chucking in a square with diagonals for every arrow, and then a cube with diagonals for every pair of such a square and arrow, and so on. Again, the proof is basically identical to the fact that that groupoidification is left adjoint to the inclusion of groupoids in categories, once more thinking of the latter in terms of intervals. It is equally obvious that this construction preserves symmetric monoidalness, again thinking of the latter in terms of a functor $C \otimes C \rightarrow C$.

7. Again thinking of monoidal categories in any flavour in terms of functors $C \otimes C \rightarrow C$, it is obvious that one can take a tensor product of two such: just form squares out of all pairs of arrows, cubes out of pairs of such squares and arrows, etc, and impose the necessary compatibilities. It is evident that the construction of 6. is the same as tensoring in this sense with the free strict cartesian category on one object.

8. One can no doubt formulate all of this quite canonically, beginning with extra structure on a monad involving intervals and cubical sets.

Posted by: anonymous on February 20, 2024 1:18 AM | Permalink | Reply to this

Post a New Comment