Topos Theory, Fall 2018

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten.

This course will be given in weeks 37--51 (Spetember-December 2018).
The course is on Mondays, 10:00-11:45 (lecture) and 12:00-12:45 (exercises), in Amsterdam Science Park, room A1.08.
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.
The hand-in exercises have to be handed in at the exercise class in the week they are mentioned below; they can also be sent by email to the teacher, before noon on the same day. Exercises, to be handed in in week x+1, will appear on this page ultimately the Wednesday of week x.
Here are the hand-in exercises with solutions for the old ones.

The course "Topos Theory" is a sequel to the course Category Theory and Topos Theory, which is prerequisite knowledge. See the lecture notes for the preliminary course.
Lecture Notes for the course (work in progress).

Making diagrams in LaTeX

Mark Kamsma prefers this tool. He gives 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 concrete:
\[
\begin{tikzcd}
Paste your code here
\end{tikzcd}
\]
Do not forget to include \usepackage{tikz-cd} in your LaTeX document to use the tikzcd environment.

Overview of the material treated in the course

• Week 37: Introduction, recap on presheaves and sheaves, synopsis of the course.
• Week 38: Beck's Crude Monadicity Theorem, Equivalence Relations and Partial Maps.
  Hand-in exercise 1. Deadline: October 1.
• Week 39: No lecture: lecturer abroad.
• Week 40: Representability of partial maps. The opposite category of a topos.
• Week 41: Beck condition; Monadicity Theorem.
  Hand-in exercise 2. Deadline: October 29.
• Week 42: Slice toposes; the Fundamental Theorem of Topos Theory; Image facorizations.
• Week 43: Introduction to Geometric Morphisms; points of presheaf toposes.
• Week 44: Flat and filtering functors; tensor products; continuous functors.
• Week 45: The topos of coalgebras for a finite-limit preserving comonad.
  Hand-in exercise 3. Deadline: November 12.
• Week 46: Surjections. The topos of sheaves for a Lawvere-Tierney topology.
  Hand-in exercise 4. Deadline: November 26.
• Week 47: Logic in toposes.
  Hand-in exercise 5. Deadline: December 10.
• Week 48: Classifying toposes.
• Week 49: Classifying toposes and geometric logic.
• Week 50: Example of a non-Grothendieck topos.
  Hand-in exercise 6. Deadline: January 18, 2019.
• Week 4: Monday January 21: Exam in building JK, Valckeniersstraat, Amsterdam, room JK1.19, 10:00--13:00.
  Here is the exam, with solutions.

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