I am reading Bell's A primer of infinitesimal analysis, and the real numbers he considers have certain properties for doing synthetic differential geometry. He calls this object the smooth real line. I am not an expert in topos theory, but know some category theory. I know that we can construct the real numbers object in a topos by Dedekind cuts or by Cauchy sequences, and that these do not always coincide in a topos. My question is Is the smooth real line the real numbers object constructed by Dedekind cuts, by Cauchy sequences in a model for synthetic differential geometry? If not, how is the smooth real line related to the other two real numbers object?
Derek (in the comments) is right: The smooth real numbers, those which potentially contain infinitesimals, do not coincide with the Dedekind or Cauchy reals.
A quick way to see this is as follows. For both the Cauchy and the Dedekind reals, it's a theorem that $x^2 = 0$ implies $x = 0$. However, this theorem contradicts the principle of microaffinity, the fundamental axiom on which SDG is built.
The exact relationship between the smooth reals and the Dedekind or Cauchy reals depends on the model under consideration. Sometimes the Dedekind reals are a quotient of the smooth reals.
(Note that the term "smooth reals" is also used for a certain subset sitting inbetween the Cauchy and the Dedekind reals. These are not the reals which make up the "smooth real line" you (and this answer) have been referring to.)