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I am trying to understand the Dedekind and Cauchy real objects in topoi concretely by looking at presheaf categories over small (tiny?) categories. For example, consider the topos which is the category of spans of sets. That is, this is the presheaf category over $\nearrow \nwarrow$. How can we construct the Dedekind and Cauchy objects concretely in this category?

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Have you tried just taking the actual construction, given in terms that make sense in any topos, and writing down what it means in that particular case? – Mike Shulman Feb 7 '12 at 21:38
To add to Mike's comment: have a guess at what the real numbers object(s) in a presheaf category might be. – Tom Leinster Feb 16 '12 at 18:15
I read from MacLane-Mordeijk that the Dedekind reals in a presheaf category is the presheaf that is constantly $R$. – user2529 Feb 25 '12 at 15:31

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