Yoneda representations of flat functors and classifying toposes (Q2919776)

In this paper, the author obtains semantic characterizations, holding for any Grothendieck site \((C,J),\) for the models of a theory classified by a topos of the form \(\mathrm{Sh}(C,J)\) in terms of the models of a theory classified by a topos \([C^{\mathrm{op}},\mathrm{Set}].\)NEWLINENEWLINEThe author exploits the natural behavior of the notion of subtopos with respect to sites to achieve an entirely semantic characterization of the models of a quotient \(T^{\prime }\) of a theory \(T\) classified by a presheaf topos \([C^{\mathrm{op}},\mathrm{Set}]\) in terms of the models of \(T\) and of the Grotendieck topology \(J\) on \(C\) defined by saying that the subtopos \(\mathrm{Sh}(C,J) \rightsquigarrow \) \([C^{\mathrm{op}},\mathrm{Set}]\) corresponds to \(T^{\prime }.\)NEWLINENEWLINEThese characterizations arise from an appropriate representation of flat functors into Grotendieck 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.NEWLINENEWLINEThis connection with classical model theory has already produced a number of useful insights.