### Category Theory and Topos Theory - 8EC

Prerequisites

Bachelor level mathematics. As this is a logic course, it is preferable that the student knows some basic definitions from logic, that is the notions of a language and of a structure for a language. Sections 2.1--2.3 from the basic logic course "Sets, Models and Proofs" (http://www.staff.science.uu.nl/~ooste110/syllabi/setsproofs15.pdf ) suffice.

Aim of the course

Familiarize students with the basic notions and techniques of Categorical Logic and Topos Theory

Category Theory was invented by S. Eilenberg and S. MacLane in 1945, with the aim of giving a precise, mathematical formulation of the idea of a "natural correspondence". An example is: the way in which to every pointed topological space corresponds a group, the fundamental group; every point-preserving continuous map between pointed topological spaces gives rise to a homomorphism between the respective fundamental groups. Category Theory came off the ground as a mathematical subject with the discovery, by Daniel Kan, of the fundamental notion of "adjunction" in 1958. Grothendieck used Category Theory for his generalization of Sheaf theory in the early 1960's, which led to the definition of (Grothendieck) topoi. Lawvere gave axioms for "elementary topoi" and stressed the fundamental importance of these structures for interpreting higher-order logic.

In this course, the students will first get familiar with basic Category Theory: categories and functors, natural transformations, limits and colimits, adjunctions, monads and cartesian closure. Applications to logic will be treated. The second part of the course will be devoted to Topos Theory: categories of presheaves and their fundamental properties; Grothendieck sites and sheaves; the associated sheaf functor theorem; the notion of "elementary topos" and the proof that presheaves and sheaves form elementary toposes; examples from logic in these structures.

If time permits, we will treat geometric morphisms and the "classifying topos theorem".

Rules about Homework / Exam

There will be a written exam. During the course, there will be 6 hand-in exercises, which count for 30% of the final grade (but this is optional, although strongly recommended). So if W is the grade for the written exam and E the grade for the exercises, the final grade will be the maximum of W and (3E+7W)/10.

Lecture Notes / Literature

Lecturer

Jaap van Oosten