TITLE: Constructive analysis and synthetic differential geometry
QUESTION [9 upvotes]: 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.

REPLY [6 votes]: 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.  You may also be interested in smooth structures on a topos.