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 $\kappa ⊲\lambda ={\lambda }^{<\lambda }$, $|\mathcal{C}|,|E|<\lambda$).