# Category Theory and Topos Theory, Spring 2018

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten. Assistant for the exercise class is Mark Kamsma (mark@markkamsma.nl).

This course will be given in weeks 6-20 (February-June 2018).
The course is on Mondays, 10:15-12:00 (lecture) and 12:15-13:00 (exercises), in Amsterdam Science Park, room A1.04.
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 assistant, 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.

Literature: Lecture Notes.
This course will treat some applications of Category Theory to Logic. Although this is only a minor part of the course, people unfamiliar with Logic may wish to consult sections 2.1-2.3 of the basic logic course Sets, Models and Proofs, for the notions of a language and structures for a language.
Suggestions for improvement of the lecture notes are very welcome!

### Making diagrams in LaTeX

I myself use the program xypic.

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

Not always there will be time enough to treat all material in the lecture. What is listed is the required reading for the exam.
• Week 6: Chapter 1: Categories and Functors, examples, special objects and arrows.
Exercises: 1,4,5,7,9.
• Week 7: Chapter 2: Natural Transformations
exercises: 20,22,27,32,33.
• Week 8: Chapter 3: Limits and Colimits
Exercises:34,35,36,38,44.
Hand-in exercise: 1
• Week 9: Chapter 4: Regular Categories and a first glimpse at Categorical Logic.
Exercises: 65,66,67,70.
• Week 10: More on Regular Categories and Regular Logic.
Exercises: 75,79,88,89,92.
Hand-in exercise: 2
Exercises: 95,98,99,103.
• Week 12: Monads and Algebras.
Exercises: 111,113,117,118,123.
Hand-in exercise: 3
• Week 13: Cartesian Closed Categories and Natural Numbers Objects.
Exercises: 125,126,130,132.
• Week 14: Easter Monday. No lecture.
• Week 15: Presheaf categories, cartesian closed structure, every object a colimit of representable presheaves, subobject classifier.
Exercises: 144,145,146,149 and: Calculate the subobject classifier in the category of presheaves over a monoid M.
Hand-in exercise: 4
• Week 16: Continuation of theory of presheaves; sections 9.1 and 9.2.
Exercises: 152,153,154,157.
• Week 17: Logic of presheaves; definition of Grothendieck topology, separated presheaves and sheaves.
Exercises: 158,160,161,162.
Hand-in exercise: 5
• Week 18: rest of 10.1, 10.2
Exercises: 164,165,167,168.
• Week 19: 10.3 and 10.4.
Exercises: 176,177.
Hand-in exercise: 6. This exercise may be sent in until May 9, midnight.
• Week 20: Further perspectives: morphisms of toposes.
Left-over exercises; questions.
• Week 25: Exam, Monday June 18, 10:00--13:00, SP C0.05.
At the exam, the use of lecture notes, books, or any kind of device which can store information, is not allowed.
Here is the exam, with solutions.
• Week 27: Retake Exam, Monday July 2, 10:00--13:00, SP B0.204.
Here is the exam of 2014, with solutions.
Here is the retake exam of 2014, without solutions.
Here is the exam of 2016, with solutions

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