TOPOS THEORY
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
A Bachelor’s degree in Mathematics (or equivalent mathematical maturity). Although some familiarity with the language of category theory and the basics of first-order logic would be desirable, no previous knowledge of these subjects is required. Indeed, the course will present all the relevant preliminaries as they are needed.
The student will be able to choose between two alternative exam modes:
- Solutions (prepared at home) to exercises assigned by the Lecturer and seminar on a suitable topic extending the contents of the course (chosen in agreement with the Lecturer). The evaluation of this seminar will constitute the two thirds of the final grade, while the remaining third will be determined by the solutions to the exercises.
- Taking two partial written exams (each of which consisting in theoretical questions and exercises), one administered half way through the course and the other immediately after the end of it.
This course is an introduction to topos theory, aimed at providing the student with an extensive theoretical preparation in this field both from a theoretical viewpoint and at the level of methodologies for effectively applying toposes in a great variety of different mathematical contexts.
At the end of the course, the student
- will have learned the central notions and results of topos theory
- will have become familiar with the basics of the interpretation of logic in categories
- will know about methods for studying mathematical theories from a topos-theoretic perspective
- will have acquired techniques for extracting new information from correspondences, dualities or equivalences and for establishing new fruitful connections between different fields.
Topos theory can be regarded as a unifying subject in Mathematics, with great relevance as a framework for systematically investigating the relationships between different mathematical theories and studying them by means of a multiplicity of different points of view. Its methods are transversal to the various fields and complementary to their own specialized techniques. In spite of their generality, the topos-theoretic techniques are liable to generate insights which would be hardly attainable otherwise and to establish deep connections that allow effective transfers of knowledge between different contexts.
The role of toposes as unifying spaces is intimately tied to their multifaceted nature; for instance, a topos can be seen as a generalized space, as a mathematical universe, but also as a theory modulo a certain notion of equivalence.
Toposes were originally introduced by Alexandre Grothendieck in the early 1960s, in order to provide a mathematical underpinning for the `exotic' cohomology theories needed in algebraic geometry. Every topological space gives rise to a topos and every topos in Grothendieck's sense can be considered as a `generalized space'.
At the end of the same decade, William Lawvere and Myles Tierney realized that the concept of Grothendieck topos also yielded an abstract notion of mathematical universe within which one could carry out most familiar set-theoretic constructions, but which also, thanks to the inherent `flexibility' of the notion of topos, could be profitably exploited to construct `new mathematical worlds' having particular properties.
A few years later, the theory of classifying toposes added a further fundamental viewpoint to the above-mentioned ones: a topos can be seen not only as a generalized space or as a mathematical universe, but also as a suitable kind of first-order theory (considered up to a general notion of equivalence of theories).
The course will start by presenting the relevant categorical and logical background and extensively illustrate these different perspectives on the notion of topos, with the final aim of providing the student with tools and methods to study mathematical theories from a topos-theoretic perspective, extract new information about correspondences, dualities or equivalences, and establish new and fruitful connections between distinct fields.
PROGRAMME:
Categorical preliminaries
Sheaves on a topological space
Sheaves on a site
Basic properties of Grothendieck toposes
Local operators
Geometric morphisms
Flat functors
Morphisms between sites
The interpretation of logic in categories
Classifying toposes and the 'bridge' technique
F. Borceux, Handbook of categorical algebra, Vols. 1-2-3, Cambridge University Press, 1994
O. Caramello, Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’, Oxford University Press, 2017
R. Goldblatt, Topoi: The Categorial Analysis of Logic, Reprint of the 1983 edition, Dover.
P. T. Johnstone, Sketches of an Elephant: a topos theory compendium. Vols. 1-2, Oxford University Press, 2002
S. Mac Lane, Categories for the working mathematician, Springer, 2nd edition, 1997
S. Mac Lane, I. Moerdijk, Sheaves in geometry and logic. A first introduction to topos theory, corrected reprint of the 1992 edition, Universitext, Springer-Verlag, New York, 1994
E. Riehl, Category theory in context, Cambridge University Press, 2016
The frontal theoretical lessons, given at the blackboard (with the support of slides), will be supplemented by sessions of exercises assigned by the Lecturer in the previous lessons, in which the students who desire to do so will be able to expose at the blackboard their solutions and discuss them with the lecturer.
The course will consist of six hours per week, of which one or two will be devoted to the discussion of exercises or specific problems.
The students of the course can reach the lecturer in her office in the hour immediately following the end of each lecture to ask for more explanations, clarifications or suggestions for further study.
Students can also take an appointment with the Lecturer by sending her an e-mail.