What is the best reference for real number objects?

Question

Are there textbooks that, assuming a decent grasp of category theory and topos theory in particular, start with a natural numbers object and build the integers, the rationals and the real numbers (dedekind and cauchy) in general and also making the explicit construction in some important toposes?

Also it should include proofs for basic theorems of real analysis valid in every (or most) topos.

Thanks!

Answer 1

There are two sections about this in Sheaves in Geometry and Logic by Mac Lane and Moerdijk:

• VI.8, which is about the construction of the real numbers object in a topos;
• VI.9, which is about Brouwer's theorem that all functions are continuous (which can hold in certain intuitionistic toposes).