The n-Category Café

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.