This is a four-hour lecture course given at IHES for the conference "Topics in Category Theory" at ICMS Edinburgh (11-13 March 2020). The videos of the course are available here.
This course provides an introduction to the theory of Grothendieck toposes from a meta-mathematical point of view. It presents the main classical approaches to the subject (namely, toposes as generalized spaces, toposes as mathematical universes and toposes as classifiers of models of first-order geometric theories) in light of the more recent perspective of toposes as unifying ‘bridges’ relating different mathematical contexts with each other and allowing to study mathematical theories from multiple points of view.This course is given jointly with Prof. Laurent Lafforgue. The lectures take place every Tuesday (14-17) and Friday (14-17) in the room VA2 in Via Valleggio 11, Como. The programme of the course is available here.
Slides used as a support for the lectures (will be regularly uploaded here):
Chapter 1: Basic features of homology and cohomology
Chapter 2: Categorical preliminaries
Chapter 3: De Rham cohomology, Poincaré duality and Lefschetz fixed points formula
Chapter 4: Sheaves, toposes and abelian categories
Chapter 5: Derived categories, derived functors and Grothendieck's six operations
Chapter 6: Čech cohomology (notes available soon)
Chapter 7: Operations on linear sheaves on sites and Grothendieck's six operations for étale cohomology
In the academic year 2018-2019 I taught (in English) a Master-level course on Topos Theory at the University of Insubria (Como, Italy).
Slides used as a support for my lectures:
Lecture 1 - Overview of the course
Lectures 2-3-4 - Categorical
preliminaries
Lectures 5-6 - Sheaves on a topological
space
Lectures 7-14 - Sheaves on a site;
basic properties of Grothendieck toposes; local operators
Lectures 15-18 - Geometric morphisms;
flat functors; morphisms between sites.
Lectures 19-20 - The interpretation of
logic in cateogories
Lectures 21-22 - Classifying toposes and
the 'bridge' technique
Students from Insubria can access the videos of the lectures at this link by inserting their credentials.
The exercises for the exam are available here.
More information about the course is available at the course webpage ; see also these presentation slides.
Click here for a(n unpolished) draft of the first two chapters of my book "Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'" (Oxford University Press, 2017). This 64 pages text is a self-contained introduction to toposes, categorical logic and the 'bridge' technique requiring only a basic familiarity with category theory.