Sh(B)-Valued Models of \((\kappa ,\kappa )\)-Coherent Categories

A basic technique in model theory is to name the elements of a model by introducing new constant symbols. We describe the analogous construction in the language of syntactic categories/sites. As an application we identify \(\textbf{Set}\)-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as "Sh(B)-valued models"). For the coherent fragment \(L_{\omega \omega }^g \subseteq L_{\omega \omega }\) this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to \(L_{\kappa \kappa }^g\) when \(\kappa \) is weakly compact. We present some further applications: first, a Sh(B)-valued completeness theorem for \(L_{\kappa \kappa }^g\) (\(\kappa \) is weakly compact), second, that \(\mathcal {C}\rightarrow \textbf{Set} \) regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.