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Proof. Consider the following diagram: Since we can assume that we have amalgamation (which is an easy consequence of inter- polation), we have that a model M is «*-saturated if and only if for every p: N— WN’ in K,,, each g: N—>M extends to some q/ : N'—> M: Theorem 3.1. Assume GCH and interpolation. Then the &* -classifying topos Set [Ts] + of the theory of K* -saturated models is precisely Sh(Kx?, Tp), where Tp is the dense topology (i.e., it is the double negation subtopos of Set* ).

Figure 1 Proof. Consider the following diagram: Since we can assume that we have amalgamation (which is an easy consequence of inter- polation), we have that a model M is «*-saturated if and only if for every p: N— WN’ in K,,, each g: N—>M extends to some q/ : N'—> M: Theorem 3.1. Assume GCH and interpolation. Then the &* -classifying topos Set [Ts] + of the theory of K* -saturated models is precisely Sh(Kx?, Tp), where Tp is the dense topology (i.e., it is the double negation subtopos of Set* ).

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By the results of [Ros97], Ks) coincides with Sat,(K). Therefore the right morphism is an isomorphism. We now want to deduce from this that f* is an equivalence, i.e., every model of size kt is Kt-saturated. Then K is «*-categorical since there is a unique such model (see [Ros97]).
By the results of [Ros97], Ks) coincides with Sat,(K). Therefore the right morphism is an isomorphism. We now want to deduce from this that f* is an equivalence, i.e., every model of size kt is Kt-saturated. Then K is «*-categorical since there is a unique such model (see [Ros97]).
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MathematicsPure MathematicsTopos theoryCategorical Latent Variable Modeling
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