Course description:
Toposes were defined by Grothendieck and his co-workers, as a generalization of categories of sheaves on topological spaces: a "Grothendieck topos" is the category of sheaves on a site, where a site is a category with a suitably defined "Grothendieck topology".
In the 1960's, Lawvere formulated the notion of an "elementary topos": a type of category characterized in purely categorical terms.
It turns out that toposes have an "internal logic", and they are useful both in geometry and logic. In the course, we shall treat the following topics: elementary toposes, geometric morphisms, inclusions and surjections, classifying toposes, logical aspects of toposes, and an example of a non-Grothendieck topos.
Prerequisites: the course Category Theory, as given in Mastermath.
Aim of the course: familiarize the students with topos-theoretic techniques, especially in Logic.
Rules about homework/exam: There will be a written exam. There will be 6 hand-in exercises during the course, which together count for 30% of the grade, provided the grade for the written exam is at least 5. Results of hand-in exercises remain valid for the retake exam.
Literature: There will be lecture notes. Further reading: MacLane and Moerdijk: Sheaves in Geometry and Logic; Johnstone: Topos Theory.
Lecturer: Jaap van Oosten (UU)
- Docent: Jaap van Oosten