On coherent topoi & coherent $1$-localic $\infty$-topoi

Abstract

In this note we prove the following useful fact that seems to be missing from the literature: the $\infty$-category of coherent ordinary topoi (in the sense of SGA4) is equivalent to the $\infty$-category of coherent $1$-localic $\infty$-topoi (in Lurie's sense). We also collect a number of examples of coherent geometric morphisms between $\infty$-topoi coming from algebraic geometry.