We obtain semantic characterizations, holding for any Grothendieck site $(C, J)$, for the models of a theory classified by a topos of the form $Sh(C,J)$ in terms of the models of a theory classified by a topos $[C^{op}, Set]$. These characterizations arise from an appropriate representation of flat functors into Grothendieck toposes based on an application of the Yoneda Lemma in conjunction with ideas from indexed category theory, and turn out to be relevant also in different contexts, in particular for addressing questions in classical Model Theory.
Keywords: Classifying topos, Yoneda lemma, flat functor, theory of presheaf type
2010 MSC: 03G30, 18C10, 18B25
Theory and Applications of Categories, Vol. 26, 2012, No. 21, pp 538-553.
Published 2012-10-12.
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