Appears in collection : Laurent Lafforgue - Classifying toposes of geometric theories

The purpose of these lectures will be to present the theory of classifying toposes of geometric theories. This theory was developped in the 1970's by Lawvere, Makkai, Reyes, Joyal and other catagory theorists, systematising some constructions of Grothendieck and his student Monique Hakim, but it still deserves to be much better known that it actually is. The last part of the lectures will present new developpments due to Olivia Caramello which, based on her principle of "toposes as bridges", make the theory of classifying toposes more applicable to concrete mathematical situations : in particular, the equivalence between geometric provability and computing on Grothendieck topologies, and general criteria for a theory to be of presheaf type.