Topos Theory, Spring 2021

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten. Teaching assistant is Bart Keller

This course will be given in weeks 6--21 (February--May 2020).
The course is on Thursdays, 14:00-15:45 (lecture) and 16:00-16:45 (exercises).
The course will be given on Teams.
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 (Bart Keller). 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 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

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 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 course

• Week 6.
  11/2/21: Introduction, preliminaries, notation; recap on presheaves on a category, sheaves on a space and etale maps.
• Week 7.
  18/2/21: Definition of geometric morphisms; Grothendieck topologies and sheaves on a site; Beck's Crude Tripleability Theorem, Adjoint Lifting Theorem. Start with Chapter 1: effectivity of equivalence relations.
• Week 8.
  25/2/21: representability of partial maps.
  Hand-in exercise 1. Deadline: March 7.
• Week 9.
  4/3/21: colimits in toposes; Beck condition; monadicity theorem.
• Week 10.
  11/3/21: slices of toposes; the "fundamental theorem of topos theory"; every topos is exact.
  Hand-in exercise 2. Deadline: March 21.
• Week 11.
  18/3/21: the structure of subobject lattices in a topos; the topos of coalgebras; if a monad T on a topos has a right adjoint, then the category of T-algebras is a topos.
• Week 12.
  25/3/21: internal categories and internal presheaves in a topos; the category of internal presheaves is a topos; Lawvere-Tierney topologies and universal closure operations; separated objects and sheaves.
  Hand-in exercise 3. Deadline: April 4.
• Week 13.
  1/4/21: Sheaves for a Lawvere-Tierney topology form a topos; geometric morphism into E.
• Week 14.
  8/4/21: introduction to geometric morphisms; points of a presheaf topos.
  Hand-in exercise 4. Deadline: April 18.
• Week 15.
  15/4/21: points of presheaf toposes (continued); flat and filtering functors; geometric morphisms from a cocomplete topos to a presheaf topos.
• Week 16.
  22/4/21: geometric morphisms from a cocomplete topos to a sheaf topos; embeddings and surjections; the factorization theorem.
• Week 17.
  29/4/21: introduction to logic in toposes.
  Hand-in exercise 5. Deadline: May 9.
• Week 18.
  6/5/21: Kripke-Joyal semantics in toposes.
• Week 19. No lecture: Ascension Day
• Week 20.
  20/5/21: Kripke-Joyal in presheaf toposes ("Kripke models") and an example; introduction to Classifying Toposes.
  Hand-in exercise 6. Deadline: May 30.
• Week 21.
  27/5/21: Last lecture. More on Classifying Toposes.
• Week 23 Exam: June 10, 14:00--17:00. The exam will be online. If you have not registered yet, but are interested in participating in the exam, send an e-mail to me.
  Here is the exam of 2019, with solutions.
  Here is the exam of 2021, with solutions.
• Week 26 Resit: July 1, 14:00--17:00. Please send me an email before June 30, if you wish to participate in the resit.

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