Olivia Caramello : “Introduction to categorical logic, classifying toposes and the 'bridge' technique” (IHES, 23 et 24 novembre 2015)



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





Lecture 1

Why logic ?

[6'00] ("digression" : Sheaves on a site, classifying toposes...)

[11'] First-order logic and its interpretation in categories. 

[31'] Fragments of logic

[38']Deduction systems

[43'] Question : universal models and functorial semantics

[48'] Categorical semantics

[1h08'] Models of a theory in a category



Lecture 2

Syntactic category associated with a theory

  Michael Makkai, Gonzalo E. Reyes :  First Order Categorical Logic,  Lecture Notes in Mathematics 611 (1977)

[48'] Universal models

[55'] Classifying toposes



Lecture 3 

(reminder) Classifying topos and universal model

Morita equivalence

[15'] Subtopos and quotient theories

[36']Lattices of theories

[57'] Theories classified by a presheaf topos and their quotients. 



Lecture 4

Zariski topos

Remarks about history of toposes

Finite presentability, irreducible formulae and homogeneous models.

The ‘bridge-building’ technique: Morita-equivalences as ‘decks’ and site characterizations as ‘arches’. 

Duality



References



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).


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