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Stream: learning: questions

Topic: Universal property of the classifying topos

Leopold Schlicht (Mar 17 2022 at 17:54):

Let $T$ be a classical first-order theory (in the sense of model theory: finitary and Boolean). Then the classifying topos $\mathcal E_T$ of $T$ is the classifying topos of the Morleyization of $T$. Does that topos have a universal property that can be formulated without referring to the Morleyization?

I think for Boolean topoi $\mathcal E$ we have $\mathrm{Geom}(\mathcal E, \mathcal E_T)\simeq T\mathrm{Mod}(\mathcal E)_e$ (the subscript $e$ indicates that the morphisms are elementary embeddings), is that correct?