Symmetric monoidal functors from powers of the natural numbers to Set

Question

Consider the full subcategory of $\mathbf{Set}$ consisting of the singleton $1$ and countable infinite sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between them interpreted as a Lawvere theory.) What are the symmetric monoidal functors from this subcategory to $\mathbf{Set}$?

Answer 1

This is not a complete answer, but I want to share my progress.

Let $\mathcal{C} \subseteq \mathbf{Set}$ be the category of sets consisting of all $\mathbb{N}^k$ with $k \in \mathbb{N}$. It is cartesian. It follows that for every symmetric monoidal category $\mathcal{D}$ the category of symmetric monoidal functors $\mathcal{C} \to \mathcal{D}$ is equivalent to the category of (finite product)-preserving functors $ \mathcal{C} \to \mathrm{CocommCoAlg}(\mathcal{D})$. If $\mathcal{D}$ is cartesian (as in your example), then the forgetful functor $\mathrm{CocommAlg}(\mathcal{D}) \to \mathcal{D}$ is an isomorphism of categories.

Thus, it suffices to find the (finite product)-preserving functors $F : \mathcal{C} \to \mathcal{D}$. (You only asked for $\mathcal{D} = \mathbf{Set}$, but I don't think that this makes the problem easier. Let us keep $\mathcal{D}$ arbitrary for now.) This essentially means to find "all the structure" on $\mathbb{N}$ in the language of cartesian categories. Namely, any structure on $\mathbb{N}$ leads to a structure on $X := F(\mathbb{N}) \in \mathcal{D}$, and clearly the object-part of $F$ is determined by $X$.

But there are actually a lot of structures. First of all, $\mathbb{N}$ carries the structure of a commutative semiring. It follows that $X$ also carries the structure of a commutative semiring. Also, for every map $f : \mathbb{N} \to \mathbb{N}$ we must have a morphism $f_* : X \to X$, and surely $\mathrm{id}_* = \mathrm{id}$ and $(fg)_* = f_* g_*$. But this also must be true for all maps $f : \mathbb{N}^k \to \mathbb{N}$, which induce morphisms $X^{k} \to X$. For example, there is an isomorphism $\mathbb{N} \cong \mathbb{N} \times \mathbb{N}$, thus inducing an isomorphism $X \cong X \times X$.

I should stop here. I don't think that there is any reasonable description.

Here is another approach, but I am not sure if it is useful. $\mathcal{C}$ is equivalent to a category $\mathcal{C}'$ which has exactly two objects, $1$ and $\mathbb{N}$. Choose an isomorphism $\delta : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ (for example, $\delta(n,m) := 2^n (2m+1) - 1$). Then $\mathbb{N} \times \mathbb{N} = \mathbb{N}$ in $\mathcal{C}'$, the projections are $p_1 \circ \delta^{-1}$ and $p_2 \circ \delta^{-1}$. The category of (finite product)-preserving functors $\mathcal{C} \to \mathcal{D}$ is equivalent to the category of (finite product)-preserving functors $F : \mathcal{C}' \to \mathcal{D}$. On objects, they are determined by $X := F(\mathbb{N})$ again. And again we have the morphisms $f_* : X \to X$ for all maps $f : \mathbb{N} \to \mathbb{N}$. And we need that $$((p_1 \circ \delta^{-1})_*,(p_2 \circ \delta^{-1})_*) : X \to X \times X$$ is an isomorphism. Probably much more is needed.

Answer 2

This was a bit too long for a comment, but I think I have essentially a complete characterization of what these things look like, up to me being unfamiliar with model theory and potentially messing something up.

First, one can let the subset of $X$ stabilized by the function which sends $1$ to itself and everything else to $0$ be $B$. This is essentially the subset of $X$ of things 'which are either $0$ or $1$'. Some fiddling around can show that $B$ is a boolean algebra, and thus must be the set of clopens on some profinite set. Each point of this profinite set is a ultrafilter on $B$, assigning a truth value to every 'predicate'. For instance, $X = 1$ has $B = 1$ and gives the empty set, $X = \mathbb N$ has $B = 2$ and gives a singleton, finite powers of $\mathbb N$ have $B = 2^k$ and give finite sets, and infinite powers gives the Stone-Čech compactification.

There is a good notion of the fiber $X_p$ of $X$ over a ultrafilter $p$ on $B$. Let $x$ and $y \in X$ have the same value at $p$ if the indicator function from $\mathbb N \cong {\mathbb N}^{\times 2}$ sending pairs $(a, b)$ to $a = b$ evaluated on $(x, y)$ is in the ultrafilter $p$.

The fiber of one such of these models is a model which assigns a definite truth value to whether $x \in X$ is supposed to be inside each subset of $\mathbb N$. Also note that each of these fibers are themselves a model of the Lawvere theory and assign to each element of $X_p$ a ultrafilter on $\mathbb N$ of the subsets that they 'belong' to.

In the case of replacing powers of $\mathbb N$ in the original Lawvere theory with the powers of some finite set $A$, this gives a complete characterization - the only possible fiber is $A$ itself and $X$ must be of the form of the maps from some profinite set to $A$.

In the case of the full natural numbers, though, there are more possibilities for the fiber. For example, a non-trivial ultrapower of $\mathbb N$ works. Note, however, that $X_p$ must be correct about all first-order sentences in the language with a symbol for every subset of $\mathbb N$. If the sentence

$$ \forall x_1, x_2 \exists y_1, y_2 \forall x_3, x_4 \exists y_3 P(x_1, x_2, y_1, y_2, x_3, x_4, y_3)$$

holds in $\mathbb N$ for a given subset $P \subset {\mathbb N}^{7}$, then one can define the functions $f_1(x_1, x_2)$, $f_2(x_1, x_2)$, and $f_3(x_1, x_2, x_3, x_4)$ sending the $x$'s to the first possible $y_i$ such that the rest of the sentence holds. Then, these functions transferred to $X$ give witness to the corresponding sentence on $X$ holding.

Conversely, functions $X^n \to X$ are a type of subset of $X^{n+1}$ and checking that they are a functions and that composition, etc. holds are first order sentences, so $X_p$'s are equivalent to such models.

This property, of being a correct model of the language extended with a symbol for every subset of $\mathbb N$, is apparently known as being a complete extension of $\mathbb N$ and it is a theorem (e.g. see Chang and Keisler's Model theory) that all complete extensions are a limit ultrapower of $\mathbb N$ itself.

A limit ultrapower of $\mathbb N$ is given by a set $I$, a ultrafilter $D$ on $I$, and a filter $V$ on $I \times I$. The elements of $\prod_{D|V} \mathbb N$ consist of $I$-indexed families of natural numbers $f$ such that the subset of $I \times I$ consisting of $\{(i, j)| f(i) = f(j)\}$ is in the filter, with two such $f$ and $g$ being equivalent if the subset of $I$ consisting of $i$ such that $f(i) = g(i)$ is in the ultrafilter $D$.

Therefore, the fibers $X_p$ must be limit ultrapowers of $\mathbb N$, so presumably $X$ must be a continuous family of limit ultrapowers over a profinite set, for whatever is the correct notion of continuity is.

If the above analysis is correct, then, it should be the case that replacing the category of all functions from powers of the naturals to itself with the category of arithmetically defined functions will give you families of models of true first order arithmetic, and replacing functions with sentences defining functions which are provably so in Peano, with two sentences being the same if they are provably such in Peano should give families of models of Peano.