The purpose of the lectures is to give a self-contained introduction to Grothendieck toposes and their presentations not only by Grothendieck sites but also by first-order geometric theories.
We will in particular focus on the notion of invariants of toposes and on their calculations in terms of sites or theories which present them.
We will also try to show how powerful are the properties of toposes and how deep, rich and multi-faceted is the theory of presentations of toposes.
The fact that any topos is associated to infinitely many sites as well as to infinitely many different theories makes possible to relate the geometric contents of sites or the semantic contents of theories (considered as syntactic objects) through their associated toposes. This is the theory of "toposes as bridges" which has been developed in the last years by Olivia Caramello. We will try to give a first introduction to this theory, illustrating it with a few examples of applications.