Category Theory and Topos Theory [Fall 2019]
This course will treat some applications of Category Theory to Logic. Although this is only a minor part of the course, people unfamiliar with Logic may wish to consult Chapters 0 and 1 of the lecture notes of Weiss and D'Mello (available from http://www.math.toronto.edu/weiss/model_theory.pdf), for the notions of a language and structures for a language.
Aims of the course:
Most mathematicians are defined by the kind of mathematical structure they study; category theorists are defined by the way they approach mathematical structure.
The guiding idea of category theory is that no mathematical structure exists on its own; it is always an individual in a broader mathematical community. The other members of this community share the same structure, and the individual is connected to them by various structure-preserving maps. In category theory one only considers those properties that can be formulated purely in terms of how individuals relate to other individuals via structure-preserving maps, thus abstracting away from any unique individual characteristics which cannot be formulated in these terms. Contrary to what one might expect, it turns out that many interesting properties can be captured this way.
Indeed, the language of category theory is both rich and useful and by now pervades modern mathematics (in particular, algebra, algebraic geometry and topology), mainly due to the influence of mathematicians like Grothendieck and MacLane. Also, it has spawned a particular approach to logic ("categorical logic").
The aim of this course is to introduce the students to the language of category theory, its main concepts (such as limits and colimits, adjoint functors, monads, Yoneda Lemma, presheaves and sheaves), and to cover the basics of categorical logic.
Benno van den Berg (University of Amsterdam)
- Docent: Benno van den Berg