Hans Schoutens, CUNY
The model-theory of categories
One could make the claim that category theory is as foundational as set-theory or model-theory. So, we should be able to transfer from one perspective to the other. In this talk, I will consider one aspect of this meta-equivalence, by introducing a theory in a very simple, one-sorted(!) language, whose models are all categories admitting a terminal object (many categories do). Many categorical constructions then turn out to be first-order. But something even more strange happens: standard categories (like the category of Abelian groups) become actually universal models! I'll explain this apparent contradiction.
In the second part of the talk, I will concentrate on one particularly interesting category: that of compact Hausdorff spaces. I will show that we can recover the natural numbers $N$ and the reals $R$, or rather, (the isomorphism classes of) their compactifications $\bar N$ and $\bar R$, by parameter-free definitions, including their order relation, addition and multiplication. Moreover, in any category that is elementary equivalent to the category of compact Hausdorff spaces, the resulting objects are then a model of PA and a real closed field respectively. Full disclosure: while I have a complete proof for the first assertion, the second is still conjectural.
Apart from some basic model-theory, category theory and topology, everything else will be explained in the talk and so it should be accessible to many.