Topos Theory, Spring 2023

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten. Teaching assistant is Samuel Klumpers

This course will be given in weeks 6--20 (February--May 2020).
The course is on thursdays, 14:00-15:45 (lecture) and 16:00-16:45 (exercises).
Lecture room: this varies per week. See planning below for the locations of the course.
The course will be concluded by a written exam, for which you must obtain at least the grade 5. There are, during the course, 6 hand-in exercises, which may count for 30% of your final grade (it is not compulsory, although strongly recommended, to hand in these exercises). That is, your final grade will be calculated as follows: let H be your grade for the hand-in exercises and W your grade for the written exam. Provided W is at least 5, your final grade will be the maximum of W and the number (3H + 7W)/10. This number has to be at least 5.5 in order to pass.
The hand-in exercises have to be sent by email to the teaching assistant (Samuel Klumpers; email s.h.e.klumpers@uu.nl). Exercises, to be handed in in week x+1, will appear on this page ultimately the thursday of week x.
Here are the hand-in exercises of last year, with solutions.

The course "Topos Theory" is a sequel to the mastermath course Category Theory, which is prerequisite knowledge. See the lecture notes for the preliminary course.
Lecture Notes for the course (work in progress).
Hand-in exercises Spring 2021, with solutions.
Hand-in exercises Spring 2022, with solutions.

Making diagrams in LaTeX

Other people prefer the tool Tikz. There are the following instructions:
Draw your diagram using that tool, and then use the button with the braces { } to copy the LaTeX code for your diagram. You can then paste the code in a tikzcd environment, or more concretely:
\[
\begin{tikzcd}
Paste your code here
\end{tikzcd}
\]
Do not forget to include \usepackage{tikz-cd} in your LaTeX document to use the tikzcd environment.

Overview of the course

• Week 6. Room MIN 0.15
  9/2/22: Introduction, preliminaries, notation; recap on presheaves on a category.
• Week 7. Room BBG077
  16/2/23: Sheaves on topological spaces; sites and sheaves.
• Week 8. Room BBG079
  23/2/23: A few categorical preliminaries (Beck's Crude Tripleability; Adjoint Lifting Theorem). Start with Chapte1 1 (Elementary Toposes): equivalence relations are effective; statement that partial maps are representable.
  Hand-in exercise 1. Deadline: March 2.
• Week 9. Room MIN0.15
  2/3/22: Proof that partial maps are representable; finish section 1.1; Proposition 1.12.
• Week 10. Room BBG 169
  9/3/23: no lecture (teacher sick).
  Hand-in exercise 2. Deadline: March 23.
• Week 11. Room BBG 169.
  16/3/23: Recap: the opposite of E; proposition 1.12; Lemma 1.13 ("Beck condition"); Theorem 1.15 (monadicity of the powerset functor: E^{op} -- E. Cor.1.16 (E has finite colimits) and start section 1.3: Slices of a topos.
• Week 12. Room BBG 169.
  23/3/23: The "Fundamental Theorem of Topos Theory": every slice of a topos is a topos, and pullback functors between slices are logical. Every topos is an exact category. Corollaries about subobject lattices.
  Hand-in exercise 3. Deadline: April 6
• Week 13. Room BBG 169.
  30/3/23: The topos of coalgebras (sketch); Eilenberg-Moore and the corollary that if T is a monad on a topos E with a right adjoint, then the T-algebras form a topos. Internal categories and internal presheaves. Lawvere-Tierney topologies and universal closure operations.
• Week 14. Room BBG 169.
  6/4/23: Category of sheaves for a Lawvere-Tierney topology forms a topos. Sheaves form an exponential ideal and are closed under finite limits. Characterizarion of separated objects and construction of sheafification functor.
  Hand-in exercise 4. Deadline: April 16.
• Week 15. Room BBG 169.
  13/4/23: Proof that the sheafification functor preserves finite limits; introduction to geometric morphisms; points of presheaf toposes.
• Week 16. Room BBG 169.
  20/4/23: Points of presheaf toposes continued; tensor product formulas for left Kan extensions; start with geometric morphisms from a cocomplete topos E to a presheaf topos.
  Hand-in exercise 5. Deadline: May 4.
• Week 17. 27/4/23: No lecture on account of King's Day.
• Week 18. Room BBG 169.
  4/5/23: Geometric morphisms from a cocomplete topos E to presheaf and sheaf categories; embeddings and surjections; the factorization theorem.
• Week 19. Room BBG 169.
  11/5/23: Logic in Toposes: Heyting algebras and Heyting structure of subobject lattices in toposes; structures for a (many-sorted) language in a topos.
• Week 20. Room BBG 169.
  18/5/23: No lecture (Ascension Day)
  Hand-in exercise 6. Deadline: June 1
• Week 21. Room BBG 169. 25/5/23: Logic in Toposes (continued): Kripke-Joyal semantics.
• Week 24. Exam: June 15, 14:00--17:00. Room Ruppert 040
• Week 27. Retake exam: July 6, 14:00-17:00. Room TBA
  Here is the exam of 2019, with solutions.
  Here is the exam of 2021, with solutions.
  Here is the exam of 2021, without solutions.
  Here is the exam of June 2023, with solutions.
  Here is the resit of July 6, 2023, with solutions.
• Week 26

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