Topos Theory, Spring 2022

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten. Teaching assistant is Freek Geerligs

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: MIN 0.09 (Minnaert Building) in weeks 6--16; BBG 315 (Buys Ballot Building) in weeks 17 and 19-20 (the thursday in week 18, May 5, is Liberation Day, a national holiday in the Netherlands).
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 (Freek Geerligs). 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).
Last year's hand-in exercises, 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 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.
  10/2/22: Introduction, preliminaries, notation; recap on presheaves on a category, sheaves on a space and etale maps.
• Week 7.
  17/2/22: 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.
  24/2/22: representability of partial maps.
  Hand-in exercise 1. Deadline: March 3.
• Week 9.
  3/3/22: colimits in toposes; Beck condition; monadicity theorem.
• Week 10.
  10/3/22: slices of toposes; the "fundamental theorem of topos theory"; every topos is exact.
  Hand-in exercise 2. Deadline: March 17.
• Week 11.
  17/3/22: 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.
  24/3/22: 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.
  31/3/22: Sheaves for a Lawvere-Tierney topology form a topos; geometric morphism into E.
• Week 14.
  7/4/22: introduction to geometric morphisms; points of a presheaf topos.
  Hand-in exercise 4. Deadline: April 18.
• Week 15.
  14/4/22: points of presheaf toposes (continued); flat and filtering functors; geometric morphisms from a cocomplete topos to a presheaf topos.
• Week 16.
  21/4/22: geometric morphisms from a cocomplete topos to a sheaf topos; embeddings and surjections; the factorization theorem.
• Week 17.
  28/4/22: introduction to logic in toposes.
  Hand-in exercise 5. Deadline: May 19.
• Week 18.
  5/5/22: no lecture (national holiday)
• Week 19.
  12/5/22: Kripke-Joyal semantics in toposes.
• Week 20.
  19/5/22: Introduction to Classifying Toposes.
  Hand-in exercise 6. Deadline: June 2.
• Week 23 Exam: June 16, 14:00--17:00. Room DDW 1.30
  Here is the exam of 2019, with solutions.
  Here is the exam of 2021, with solutions.
• Week 26 Resit: July 7, 14:00--17:00. Room BBG 017. Please send me an e-mail before June 30, if you wish to participate in the resit.

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