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1. Idea
Recall that it is possible to define an internalization of the set of natural numbers, called a natural numbers object (NNO), in any cartesian monoidal category (a category with finite products). In particular, the notion makes sense in a topos. But a topos supports intuitionistic higher-order logic, so once we have an NNO, it is also possible to repeat the usual construction of the integers, the rationals, and then finally the real numbers; we thus obtain an internalization of R in any topos with an NNO.

More generally, we can define a real numbers object (RNO) in any category with sufficient structure (somewhere between a cartesian monoidal category and a topos). Then we can prove that an RNO exists in any topos with an NNO (and in some other situations).

2. Definition
Let E be a Heyting category. (This means, in particular, that we can interpret full first-order intuitionistic logic using the stack semantics.)

5. Examples
In Set
The real numbers object in Set is the real line, the usual set of (located Dedekind) real numbers. Note that this is a theorem of constructive mathematics, as long as we assume that Set is an elementary topos with an NNO (or more generally a Π-pretopos with NNO and either WCC or subset collection).

In sheaves on a topological space

Thus, for every topological space X, the topos Sh(X) has a Dedekind real numbers object R. Naively one might expect R to be isomorphic to the constant sheaf Δ(R), where R is the classical set of real numbers, but this turns out not to be the case. Instead, we have a rather more remarkable result:

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