For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject classifier, by taking finite products, finite coproducts, exponentials (internal homs) and taking subobjects defined by [edit 2021-02-16] finitary formulas in the internal language (using previously defined objects and relations). (I'm saying this a bit concisely in the hope that there are no major subtleties.)

In particular, if $X$ is a topological space and we consider the topos of sheaves on $X$, the sheaf of continuous functions with values in each one of the following is definable (along with its usual algebraic structure):

$\mathbb{N}$ with the discrete topology (this is the natural numbers object),

$\mathbb{Z}$ with the discrete topology (this is the Grothendieck group of the previous),

$\mathbb{Q}$ with the discrete topology (this is the fraction field of the previous),

$\mathbb{R}$ with the usual (Euclidean) topology (by Dedekind cuts).

Let us call $\mathbf{N}, \mathbf{Z}, \mathbf{Q}, \mathbf{R}$ the corresponding definable objects of the topos.

Now for a long time I thought one could not define the sheaf of continuous functions with values in

- $\mathbb{Q}$ with the usual (i.e. induced by $\mathbb{R}$) topology,

but I serendipitously realized that you can, namely it is given by the following object:

$$\{x\in\mathbf{R} : \forall y\in\mathbf{R}. ((\forall z\in\mathbf{Q}.(y\mathrel{\#}z)) \Rightarrow (x\mathrel{\#}y))\}$$

where $x\mathrel{\#}y$ stands for $(x<y)\lor(x>y)$ or, equivalently, $\exists z\in\mathbf{R}.(z\cdot(x-y)=1)$.

(This is easy to see: first note that $\{x\in\mathbf{R} : \forall y\in\mathbf{Q}. (x\mathrel{\#}y)\}$ defines the sheaf of continuous functions with values in $\mathbb{R}\setminus\mathbb{Q}$ with the usual topology, then repeat the reasoning.)

So now I am curious to know whether the “converse” is possible:

- $\mathbb{R}$ with the discrete topology;

in other words:

**Question:** is the sheaf of locally constant real-valued functions on $X$ definable, uniformly in $X$, as a subobject of $\mathbf{R}$ in the topos of sheaves on $X$?

I imagine there is little hope of finding a good answer to the very general question “for which topological spaces $Y$ is the sheaf of continuous $Y$-valued functions on $X$ definable as an object in the topos?”, but of course, if someone wants a crack at it rather than the particular case above, by all means do!

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