I am curious if (any of) the various inequivalent constructions of the real line in constructive mathematics can be used to build a model of Kock and Lawvere's synthetic differential geometry? In other words, do any of the constructions of the real line (in say HoTT) satisfy the Kock-Lawvere axiom for a class of functions which deserve to be called smooth? If not, how can we "augment" the real line with nilpotent infinitesimals for this to be true? 


I am a complete novice when it comes to constructive mathematics, but I'm reasonably comfortable with Anders Kock's synthetic differential geometry texts. Unfortunately, I haven't had a chance to read the Models of SDG text yet, so I apologize if this is covered there.



Constructive analysis and synthetic differential geometry
In the smooth-topos models of SDG, the situation is generally something like this. The internally-definable Cauchy real numbers $\mathbf{R}\_c$ are the sheaf of locally constant $\mathbb{R}$-valued functions, while the internally-definable Dedekind real numbers $\mathbf{R}\_d$ are the sheaf of continuous $\mathbb{R}$-valued functions. In between these there is the sheaf $\mathbf{R}\_s$ of *smooth* $\mathbb{R}$-valued functions, $\mathbf{R}\_c \to \mathbf{R}\_s \to \mathbf{R}\_d$, which in turn is a quotient of the "line object" $\mathbf{R}\_l$ that contains infinitesimals. Despite much trying, I don't know of any internal construction that produces $\mathbf{R}\_s$ or $\mathbf{R}\_l$ in these toposes; it seems that they have to be considered extra structure with which the topos is equipped. However, they are related to other structure it has, such as [differential cohesion](https://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos). You may also be interested in [smooth structures on a topos](https://ncatlab.org/nlab/show/smooth+structure+on+a+topos).


