How to construct the Dedekind and Cauchy real objects in the topos of spans of sets?

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?

flag	asked Feb 3 2012 at 14:31 Colin Tan 4,211●13●43

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 2012 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 2012 at 18:15

I read from MacLane-Mordeijk that the Dedekind reals in a presheaf category is the presheaf that is constantly $R$. – Colin Tan Feb 25 2012 at 15:31

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