Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?

Question

The question is in the title, here is my motivation:

$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if $\mathcal C$ has infinite coproducts, the object $N:=\displaystyle\coprod_{n\in\mathbb N}I$ has the following properties :

• it is a natural numbers object (NNO) in the sense of Lawvere, ie it has natural morphisms $0:I\to N$ and $S:N \to N$ such that any given $I\to X\to X$ uniquely defines a morphism $f:N\to X$ such that the following diagram commutes

$$\begin{CD} I @>0>> N @>S>> N\\ @| @VfVV @VfVV\\ I @>>> X @>>> X \end{CD}$$

• it is also the free monoid on $I$, ie the initial object in the category of monoids under $I$

If such a coproduct does not exist, a natural numbers object (if it exists) is the free monoid on $I$, but the converse is not necessarily true (or at least, I haven't been able to prove it). To better understand what might happen, It would help to see a workable example of such a category.

Of course for such an example to be interesting, I would like to have a category where an NNO or the free monoid exists.

Answer 1

The effective topos is an example of a (locally!) cartesian closed category with NNO where infinite coproducts do not exist. Indeed, the main feature of the NNO $N$ in the effective topos is that the endomorphisms of $N$ are precisely the computable functions. As such, $N$ cannot be (isomorphic to) $\coprod_{n \in \mathbb{N}} 1$: if that were the case, then there would be uncountably many endomorphisms of $N$. (Of course, $\coprod_{n \in \mathbb{N}} 1$ does not even exist in the effective topos; as you say, if it existed, it would have to be the NNO.)

Here is a closely related construction. Suppose we have a countable model $M$ of Zermelo set theory, or even just Mac Lane set theory. (For instance, by Skolem, we may take $M$ to be a countable elementary substructure of $V_{\omega + \omega}$.) Then the category of sets in $M$ is an elementary topos (so locally cartesian closed) with NNO, but its NNO cannot be $\coprod_{n \in \mathbb{N}} 1$, for the same reason.

Answer 2

Another example is a filterquotient topos. E.g. let $\mathcal{U}$ be a nonprincipal ultrafilter on $\mathbb{N}$ and consider the filterquotient of $Set/\mathbb{N}$ by $\mathcal{U}$. This is a world of "nonstandard analysis" which has a NNO (the image of $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$) but not every external infinite sequence of elements is an internal one, so the NNO is not $\coprod_{n\in\mathbb{N}} 1$.