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

Abstract 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 $$\textbf{Set}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Set</mml:mi> </mml:math> -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 $$L_{\omega \omega }^g \subseteq L_{\omega \omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msubsup> <mml:mo>⊆</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to $$L_{\kappa \kappa }^g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mi>κ</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msubsup> </mml:math> when $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> is weakly compact. We present some further applications: first, a Sh ( B )-valued completeness theorem for $$L_{\kappa \kappa }^g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mi>κ</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msubsup> </mml:math> ( $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> is weakly compact), second, that $$\mathcal {C}\rightarrow \textbf{Set} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Set</mml:mi> </mml:mrow> </mml:math> regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.