Introduction to categorical logic, classifying toposes and the 'bridge' technique The course will begin by presenting the basic notions and results of first-order categorical logic, with the aim of reaching the theory of classifying toposes by Makkai and Reyes and illustrating the general techniques allowing to use them as unifying 'bridges' for transferring information across distinct mathematical theories. The exposition will be accompanied by several examples and applications. The lectures will require a basic familiarity with the fundamental notions of topos theory, as reviewed in André Joyal's lectures on Monday. Lecture 1: First-order logic and its interpretation in categories. Geometric theories and syntactic categories. Universal models and representability.