In every smooth topos there is a notion of infinitesimal object and of infinitesimal number. The most common such infinitesimal numbers are nilpotent, but in some special smooth toposes, there is in addition a notion of invertible infinitesimal. In these toposes there is likewise an object of smooth natural numbers, which contains infinite or nonstandard natural numbers? (and whose inverses are invertible infinitesimals).
Some examples of such smooth toposes are discussed at Models for Smooth Infinitesimal Analysis.
The phenomenon of “smooth” nonstandard natural numbers in a Grothendieck topos arises from the following simple general principle:
Consider any sheaf topos $\mathcal{T} = Sh(C)$ such that
the category Set embeds full and faithfully into $\mathcal{T}$.
(For smooth toposes we have, if they are “well adapted”, even a full and faithful inclusion of Diff and then the one of Set is the one induced by the inclusion $Set \hookrightarrow Diff$.)
the Grothendieck topology on $C$ is on each object given by finite covering families.
In such a case there are two objects in $\mathcal{T}$ that both look like they should qualify as the internal object of natural number, but that are different:
The image $N$ of the set $\mathbb{N}$ under the given full and faithful embedding.
This yields, trivially, a sheaf such that morphisms from any other set into it are given by arbitrary $\mathbb{N}$-valued functions on this set.
The abstractly defined natural numbers object $\Delta(\mathbb{N})$:
this is the sheafification of the presheaf that is constant on the set $\mathbb{N}$. A morphism into this presheaf is a constant $\mathbb{N}$-valued function. And since we are sheafifying, by assumption, with respect to finite covers, a morphism from a set into its sheafification is a function into $\mathbb{N}$ that is constant on each patch of a finite cover of that set and hence is a bounded $\mathbb{N}$-valued function.
The unbounded functions thus represent infinite? or non-standard? “smooth natural numbers.” In particular, a generalized element $n \in \Delta(\mathbb{N})$ with domain of definition $\mathbb{N}$ (regarded as an object of $\mathcal{T}$) is a bounded sequence of integers, whereas a similarly defined generalized element $\nu \in N$ is a possibly unbounded sequence of integers. This is intuitively similar to the unbounded sequences of numbers that represent infinitely large numbers in the ultrafilter approach to nonstandard analysis (a different way of making infinitesimal numbers precise).
The generic non-standard natural number is the generalized element of $N$ on the domain of definition $\ell C^\infty(\mathbb{N})/{NullTail}$ given by the canonical injection $\ell C^\infty(\mathbb{N})/NullTail \to N$ that is dual to the canonical projection of the ring onto its quotient. Here $NullTail$ is the ideal of sequences of real numbers that vanish above some integer.
The ring $C^\infty(\mathbb{N})/NullTail$ here is a quotient ring of sequences as above, where two sequences are identified if they agree above some integer. So $\ell C^\infty(\mathbb{N})/NullTail$ is the smooth locus whose function algebra is similar to a nonstandard extension of $\mathbb{R}$.
The generic non-standard natural number is discussed on page 252 of the Moerdijk-Reyes book below.
See in
chapter VI – there section 1.6 section 2 – and chapter VII.