## Time and Place:

Wednesday 16:15-18:00 in the MPI Seminar Room

## Instructor:

David Carchedi, Max Planck Institute for Mathematics, office: 314

Wednesday 16:15-18:00 in the MPI Seminar Room

David Carchedi, Max Planck Institute for Mathematics, office: 314

There will be homework problems each week. Each class session shall start with a lecture, a short break, and then a session dedicated to the previous week's homework problems. The homework problems will appear on this site.

Topos theory has many different guises. On one hand, a Grothendieck topos is a generalization (in fact categorification) of a topological space, a viewpoint which underpinned Grothendieck's own intuition on topoi, and aided his proof of one of the Weil conjectures. On the other hand, every topos can be thought of as a mathematical universe itself in which one can do mathematics. In fact, there is a duality between Grothendieck topoi and certain first-order theories of logic, called geometric theories. The interplay of these many different sides of topos theory is what makes the subject so rich.

As a first introduction to this subject, this course will take a geometric emphasis, spending more time developing the role of topoi as generalized spaces, and as universes of generalized spaces.

- Intro
- Lecture 1: Presheaf categories and the Yoneda lemma
- Lecture 2: Sheaves on spaces
- Lecture 2.5: Every presheaf is a colimit of representables
- Lecture 3: Stalks and the étalé space construction
- Lecture 4: The étalé space construction continued
- Lecture 5: Sheaves on a category of spaces and Grothendieck universes
- Lecture 6: Grothendieck topologies.
- Lecture 7: The "plus construction"
- Lecture 8: Categorical properties of topoi and Giraud's theorem.
- Lecture 9: Frames, locales, and Stone duality
- Lecture 10: Proof of Giraud's theorem
- Lecture 11: Geometric morphisms
- Lecture 12: Diaconescu's cover theorem, geometric theories, and classifying topoi

Students wishing to take this course should be familiar with some basic concepts from category theory, e.g. natural transformations, limits and colimits, adjunctions etc. Familiarity with 2-categories/bicategories is helpful, but not at all necessary.

The following book is an excellent introduction to topos theory:

It covers many (but not all) topics of this course in a greater detail than possible in these lectures (and also many other interesting topics). It is a very instructive read.

A much larger and even more exhaustive source is:

This reference is not recommended to learn the subject, as it is too vast, but it is quite useful to use as a reference to look up definitions.

We also highly recommend:

for a very formally categorical introduction to both topos theory and the theory of locales.

Finally, here is a link to slides from a course on topos theory (with a slightly different emphasis than this course) given by Olivia Caramello: