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.
O.Caramello
[6'00] ("digression" : Sheaves on a site, classifying toposes...)
[43'] Question : universal models and functorial semantics
[1h08'] Models of a theory in a category
Michael Makkai, Gonzalo E. Reyes : First Order Categorical Logic, Lecture Notes in Mathematics 611 (1977)
O. Caramello, Topos-theoretic background (and the papers cited therein), available at http://preprints.ihes.fr/2014/M/M-14-27.pdf (2014), to appear in a forthcoming book for Oxford University Press.
P. T. Johnstone, Sketches of an Elephant: a topos theory compendium, Vols.1 and 2, Oxford University Press (2002).
S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic, Springer Universitext (1992).
M. Makkai and G. Reyes, First-order categorical logic, Lecture Notes in Mathematics, Vol. 611, Springer-Verlag (1977).