Category Theory and Topos Theory, Spring 2014

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten.

This course will be given in weeks 6-21 (February-June 2013).
The course is on Mondays, 10:15-13:00, in Amsterdam Science Park C1.112 (except for Week 13, when it will be F1.02)
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 lecturer, before noon on the same day.
Here are the hand-in exercises with solutions for the old ones.

Literature: Lecture Notes for Category Theory.
Lecture Notes for Topos Theory.

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.
• Week 8: Chapter 3: Limits and Colimits
  Exercises:33,36,37,38,56
  Hand-in exercise: 1
• Week 9: Chapter 4: Regular Categories and a first glimpse at Categorical Logic.
  Exercises: 64,68,69.
• Week 10: More on Regular Categories and Regular Logic; definition of adjunctions.
  Exercises: 85,86,88,90,92,94.
• Week 11: Adjunctions.
  Exercises: 98,103,106,107.
  Hand-in exercise: 2
• Week 12: Monads and Algebras.
  Exercises: 107,112,115,120.
• Week 13: Cartesian Closed Categories and Natural Numbers Objects.
  Exercises: 121,122,123,126.
  Lecture in F1.02
  Hand-in exercise: 3
• Week 14: Presheaf categories, cartesian closed structure, every object a colimit of representable presheaves, subobject classifier.
  Exercises: 1,2 and: Calculate the subobject classifier in the category of presheaves over a monoid M.
• Week 15: Continuation of theory of presheaves; sections 1.1 and 1.2.
  Exercises: 3,4,5,6,7.
  Hand-in exercise: 4
• Week 16: Logic of presheaves; definition of Grothendieck topology, separated presheaves and sheaves.
  Exercises: 8,9,13(Note: here you have to assume that K has a top element!),14,16,19
• Week 17: Easter Monday: no lecture
• Week 18: rest of 2.1, 2.2
  Exercises: 18,21,22,23.
  Hand-in exercise: 5
• Week 19: May 5, National Holiday: no lecture
• Week 20: 2.4
  Exercises: 25,26,27,29,30
  Hand-in exercise: 6
• Week 21: Further perspectives in Topos Theory: geometric morphisms, classifying toposes, topological applications. No exercises.
• Week 22: Exam Monday May 26, 10:00--13:00, OMHP D1.09
  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 25: Retake exam Monday June 16, 10:00--13:00, Science Park B0.204.
  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.

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