Every theory is eventually of presheaf type

Abstract

We give a detailed and self-contained introduction to the theory of $\lambda $-toposes and prove the following: 1) A $\lambda $-separable $\lambda $-topos has enough $\lambda $-points. 2) The classifying $\lambda $-topos of a $\kappa $-site $(\mathcal{C},E)$ is a presheaf topos (assuming $\kappa \vartriangleleft \lambda =\lambda ^{<\lambda }$, $|\mathcal{C}|,|E|<\lambda $).