Classifying topos

Summary

A classifying topos of a geometric theory $\mathbb{T}$ is a Grothendieck topos that contains a generic model of $\mathbb{T}$. Models of $\mathbb{T}$ in any Grothendieck topos $\mathcal{E}$ correspond to geometric morphisms from $\mathcal{E}$ to the classifying topos. Every geometric theory has a classifying topos, unique up to equivalence. This turns questions about theories into questions about toposes, and lets one read a topos as a theory.