Course syllabus

Welcome to Category Theory, LOG350, 7,5 credits

This course is both part of the Master's Programme in Logic (Links to an external site.) as well as available as a free standing course. 

The course syllabus is available here: course syllabus (Links to an external site.)

Course content

The course starts with general category theory and defines the concept of a category. Examples of categories, constructed in set theory, are presented. Then, a number of central concepts in category theory, defined by using abstract limits and universal properties, are presented. The course also provides an introduction to topos theory, and its connection to logic and set theory. Finally, two central concepts in general category theory are defined and exemplified: functors and natural transformations.

Teachers

The course will be taught by Paul Kindvall Gorbow. You can contact Paul by email, paul.gorbow@gu.se.

Introduction

The introduction to the course will take place on September 6, 9:15 at Olof Wijksgatan 6. The room is T304.

Registration

You can find information regarding registration here (Links to an external site.).

Schedule

The schedule for the course is available through TimeEdit.

Literature

The course text book is "Topoi: The Categorial Analysis of Logic" by Robert Goldblatt. Chapter pdf:s are freely available through Project Euclid.

Examination

The course is assessed individually in the form of oral student presentations and written home assignments.

Grading

The grade is based on the problem set solutions and the seminar.

For Problem sets, above 66% score is G, and above 80% score is VG. If you have below 66% on a problem set, you can hand it in again. If so, I can give hints. All problem sets need to be passed to pass the class.

The seminar will primarily be assessed on clarity of presentation. The point of the seminar is to convey the topic to the audience. To pass it is sufficient to make a good effort; it doesn't matter if you make some mistakes, or get confused. The difficulty of the topic is not important to the grade.

A total of at least 80% on problem sets (first try) and a reasonably clear presentation will yield a VG grade. With a good presentation, less than 80% on problem sets can also give VG.

An alternative path to a passing grade (G) is also offered, to accommodate students with a non-mathematical background who find that independent problem solution is too difficult: This consists of writing an essay summarizing the topics of the course and giving an oral exam for me, solving 3 problems that can be prepared in advance. (No need to solve the problem set and make a seminar.) Please contact me for setting up the details if you want to choose this alternative.

Suggested seminar topics

Each student is required to make a seminar. The seminar needs to be 50 min. If someone wants to present a larger topic, a longer seminar (up to 100 min) is also great, but it needs to be declared to me in advance. It's good to leave some time for questions, during or after the presentation.

It is your responsibility to choose a topic that you feel comfortable that you will manage, but I'm happy to discuss if you are unsure. It doesn't need to be advanced. That said, if you are yearning to learn some advanced topic, presenting it is a great way to do so and I won't stand in your way.

The presentation should be clear, and focused on conveying the topic to the audience. It should include some rigorous definitions and proof of a result, or carefully working through examples. Apart from that, you are quite free. It may full of formal proofs, or more on the level of a survey. You can have slides, or do it on the board. It can be a monologue with questions at the end, or more of a dialogue with the audience. It can be in the style of teaching, or in the style of a research presentation. You may do a presentation that presumes more background knowledge than what everybody in the class has. But I'll try to arrange seminars so that the background has been covered beforehand.

Here are some suggestions for topics. I have also included my own subjective assessment of the level of difficulty of the topic as [n], on a scale 1 <= n <= 5,  where n=1 means easy and n=5 difficult. This is just a very rough guide, a topic can naturally be treated in a more or less advanced way.

• Any section or sections from the book covered in the lecture plan below. I kind of recommend this, for example to choose a couple of ch. 3 sections, since it avoids getting too advanced. (But for a ch. 3 topic, you would need to start preparing relatively quickly to fit the lecture plan.)
  • Ch. 3 [1-2]
  • Ch. 4.1-4.4 [2]
  • Ch 4.6-4.8 [3-4]
  • Ch. 5 [3-4]
  • Ch. 9.1-9.2 [2-3]
  • Ch. 9.4 [4]
• The Curry-Howard correspondence [3]
• Bundles and sheaves (ch. 4.5) [4]
• Natural numbers objects (ch. 12.2) [4]
• Adjunction [3]
• The Yoneda lemma [4]
• Monads [4]
• Monoidal categories [4]
• Zoom in on a particular category. On an easy level, work out what some of the limits are in the category (or that some limit doesn't exist); on an advanced level, check out the page on nLab. Examples:
  • The category Rel of sets with relations as morphisms [2-3]
  • The category Cat of categories with functors as morphisms [3-4]
  • The category Grp of groups [2-4]

Weekly lecture plan

(This plan will be updated as the course goes along, with student seminars and perhaps some additional lecture by me.)

The chapters specified are the readings to be done before class.

5/9 - Introduction: Structures and homomorphisms

10/9 - Categories (ch. 1-2)

12/9 - Limits (ch. 3.1-3.6)

17/9 - Limits (ch. 3.7-3.10)

19/9 - Limits (ch. 3.10-3.14) - Alexander's seminar Deadline for choosing a seminar topic

24/9 - Limits (ch. 3.15-3.16)

26/9 - Topoi (ch. 4.1-4.3)

1/10 - Topoi (ch. 4.4, 4.6) Hand in: Problem set 1

3/10 - Topoi (ch. 4.7-4.8)

8/10 - Functors (ch. 9.1-9.2)

10/10 - Functors (ch. 9.3) & The Yoneda Lemma - Elena's seminar

15/10 - Topos structure (ch. 5) - Carlos's, Fabian's and/or Irene's seminars

17/10 - Topos structure (ch. 5) - Carlos's, Fabian's or Irene's seminar & Logic in topoi  Hand in: Problem set 2

22/10 - Logic in topoi & Monoidal categories - Warrick's seminar

24/10 - Categorial set theory (part of ch. 12) - Tjeerd's seminar & Concluding remarks & Course evaluation

Learning outcomes

See the course syllabus (Links to an external site.) for more information.

Special pedagogical support

If you have a disability and are in need of special pedagogical support please see the information available at the student portal (Links to an external site.). 

Contact information

• Course coordinator Paul Kindvall Gorbow, paul.gorbow@gu.se, answers questions about the course content, literature and schedule.
• Education administrator Linda Aronsson, linda.aronsson@gu.se answers questions about registration, examination administration, study interruptions, study breaks, certificates, etc. 
• Masters Program Coordinator Fredrik Engström, fredrik.engstrom@gu.se is responsible for programme issues and study guidance for students of the programme.
• Student counselor Peter Johnsen, peter.johnsen@gu.se, is responsible for study guidance of the free-standing course.