Terence Tao's cheap non-standard analysis can be interpreted as happening in a certain elementary topos, which I define here. Amusingly, this construction can itself be interpreted as happening in the category of sheaves on a certain space, but I digress.

Let $A$ be a set and let $F$ be a filter of $P (A)$. We define $\textbf{Set}^A / F$ to be the following category:

• The objects are indexed families of sets where the indexing set is in $F$.

• The morphisms $X \to Y$ are equivalence classes of indexed families of maps $f$ where:

  • The indexing set is in $F$.
  • For each index $a$ where $f_a$ is defined, $X_a$ and $Y_a$ are also defined and $f_a$ is a map $X_a \to Y_a$.
  • Two families are equivalent if the set of indices where they coincide is in $F$.

• Composition and identities are inherited from $\textbf{Set}$.

Then $\textbf{Set}^A / F$ is an elementary topos: finite limits and power objects can basically be constructed componentwise. In particular, the diagonal embedding $\Delta : \textbf{Set} \to \textbf{Set}^A / F$ is a logical functor. Since natural numbers objects are preserved by logical functors, it follows that $\textbf{Set}^A / F$ has a natural numbers object: $\Delta (\mathbb{N})$. Furthermore, $\textbf{Set}^A / F$ is boolean and satisfies the external axiom of choice. Thus, $\textbf{Set}^A / F$ satisfies the axioms of ETCS except possibly for well-pointedness.

It is not hard to see that the lattice of subobjects of $1$ in $\textbf{Set}^A / F$ is isomorphic to the boolean algebra $P (A) / F$. In particular, if $F$ is not an ultrafilter, then $\textbf{Set}^A / F$ is not well-pointed: it will have subterminal objects that are neither $0$ nor $1$. Conversely, if $F$ is an ultrafilter then $\textbf{Set}^A / F$ is well-pointed – so we obtain a non-standard model of ETCS.

In general, $\Delta : \textbf{Set} \to \textbf{Set}^A / F$ fails to have a right adjoint. Indeed, if $F$ is any non-principal filter, then $\Delta (F)$ is not the coproduct of its standard elements. Let $\mu_a : F \to 2$ be defined as follows: $$\mu_a (B) = \begin{cases} 0 & \text{if } a \notin B \\ 1 & \text{if } a \in B \end{cases}$$ Then, for each $B \in F$, $\mu (B)$ is equal to $1$ as morphisms $\Delta (1) \to \Delta (2)$. On the other hand, $\{ a \in A : \mu_a = 1 \} = \bigcap_{B \in F} B \notin F$, so $\mu$ is distinct from the constant $1$ as morphisms $\Delta (F) \to \Delta (2)$. Hence, the standard elements $\Delta (1) \to \Delta (F)$ do not comprise a jointly epimorphic cocone and cannot be a coproduct cocone.

(I think $\textbf{Set}^A / F$ also fails to be cocomplete in general, but the argument escapes me. Certainly this is so when $F$ is not closed under countable intersections – then $\Delta (\mathbb{N})$ fails to be the coproduct of its standard elements, but in an elementary topos, if the coproduct of countably many copies of $1$ exists, then the natural numbers object is the coproduct of the standard numerals.)