Category Theory and Topos Theory, Spring 2016

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten. Assistant for the exercise class is Anton Golov (a.m.golov@students.uu.nl).

This course will be given in weeks 6-21 (February-June 2016).
The course is on Mondays, 10:15-12:00 (lecture) and 12:15-13:00 (exercises), in Amsterdam Science Park F1.02. Important News: as of March 7, all lectures will be in G4.15!
The course will be concluded by a written exam. 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. 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 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.

Overview of the material treated in the course

• 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
• Week 11: Adjunctions.
  Exercises: 95,98,99,103.
• Week 12: Monads and Algebras.
  Exercises: 111,113,117,118,123.
  Hand-in exercise: 3
• Week 13: Easter Monday: no lecture
• Week 14: Cartesian Closed Categories and Natural Numbers Objects.
  Exercises: 125,126,130,132.
  Hand-in exercise: 4
• 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.
• Week 16: Continuation of theory of presheaves; sections 9.1 and 9.2.
  Exercises: 152,153,154,157.
  Hand-in exercise: 5
• Week 17: Logic of presheaves; definition of Grothendieck topology, separated presheaves and sheaves.
  Exercises: 158,160,161,162.
• 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 the evening of May 11. If you downloaded the exercise on May 4 before 18:30, please check the new version!
• Week 20: Whitsun Monday: no lecture
• Week 21: Further perspectives: morphisms of toposes.
  No exercises.
• Week 22: Exam Monday May 30, 10:00--13:00, REC M 1.01 (Plantage Muidergracht 12)
  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 of 2014, with solutions.
  Here is the exam of 2016, with solutions
• Week 25: Retake exam Monday June 20, 10:00--13:00, Science Park G3.02
  Registration for the retake exam is required! Send an e-mail to me, j.vanoosten@uu.nl, before June 10, if you wish to take part in the retake exam.
  Here is the retake exam of 2014, without solutions.

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