Topos Theory, Spring 2024

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten. Teaching assistant is Umberto Tarantino.

This course will be given in weeks 6--21 (February--May 2024).
The course is on thursdays, 14:00-15:45 (lecture) and 16:00-16:45 (exercises).
Lecture room: BBG005 (Buys Ballotgebouw)
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 (Umberto Tarantino; email umberto.tarantino@studenti.unimi.it). 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.
  8/2/24: Chapter 1. Notational conventions; monadicity and creation of limits; the Crude Tripleability Theorem; Adjoint Lifting Theorem; Eilenberg-Moore Theorem; regular and exact categories
• Week 7.
  15/2/24: Chapter 2. Categories of presheaves; universal property of category of presheaves.
• Week 8.
  22/2/24: Chapter 2. Topos structure on categories of presheaves. Examples. Cauchy-complete categories and how to recover C from its category of presheaves.
  Hand-in exercise 1. Deadline: February 29.
• Week 9.
  29/2/24: Section 3.1. Elementaary toposes: definition. The singleton map {.}. Equivalence relations and partial maps.
• Week 10.
  7/3/24: Section 3.2 and part of 3.3. E^{op} for a topos E. Monadicity Theorem for toposes. A topos has finite colimits. Structure of slices of toposes.
  Hand-in exercise 2. Deadline: March 14.
• Week 11.
  14/3/24: Sections 3.3 and 3.4. Geometric morphisms between slice toposes. The initial object is strict and coprojections are monic in a topos. Topos of coalgebras for a finite-limit preserving comonad.
• Week 12.
  21/3/24: Section 3.5 and part of 3.6. Internal categories and internal presheaves. Lawvere-Tierney topologies; universal closure operations; category of sheaves for a Lawvere-Tierney topology.
  Hand-in exercise 3. Deadline: March 28
• Week 13.
  28/3/24: Sh_j(E) is a topos; the inclusion into E has a left adjoint which preserves finite limits. Examples of Lawvere-Tierney topologies. Grothendieck topologies on a small category.
• Week 14.
  4/4/24: Sheaves on a space: local homeomorphisms and local sections. Definition and examples of geometric morphisms. Points of presheaf toposes.
  Hand-in exercise 4. Deadline: April 11.
• Week 15.
  11/4/24: Flat and filtering functors. Geometric morphisms E-->\hat{C} for arbitrary cocomplete E; geometric morphisms E-->Sh_j(C, Cov). Surjections and embeddings; factorization theorem.
• Week 16.
  18/4/24: Examples of factorizations. Characterization of embeddings. Logic in toposes: the Heyting algebra structure on subobject lattices.
  Hand-in exercise 5. Deadline: April 25.
• Week 17.
  25/4/24: Quantifiers; first-order structures in toposes; Kripke-Joyal semantics.
• Week 18.
  2/5/24: Applications of Kripke-Joyal. Kripke-Joyal in presheaf and sheaf toposes. Natural numbers in toposes.
  Hand-in exercise 6. Deadline: May 21.
• Week 19.
  9/5/24: No lecture (Ascension Day).
• Week 20.
  16/5/24: A model for "every real-valued function on the reals is continuous". Examples of classifying toposes.
• Week 21. 23/5/24: Definition of classifying toposes. Geometric logic and statement of the classifying topos theorem.
• Week 24. Exam: June 13, 14:00--17:00. Room BBG 065
• Week 27. Retake exam: July 4, 14:00-17:00. Room BBG 077
  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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