Category Theory, Fall 2020

Meeting schedule

We meet every Wednesday from 10:00-11:00 in Lille Auditorie, Incuba.
Meeting Date Topic Reading Most relevant exercises
01 Wed Sep 02 Categories SA Ch. 1 SA Ch. 1: 1, 2, 3, 5, 6, 7, 11
02 Wed Sep 09 Abstract Structure SA Ch. 2 SA Ch. 2: 1-5, 13-15, 17, 18
03 Wed Sep 16 Duality SA Ch. 3 SA Ch. 3: 1-4, 6(monoids), 10-14
04 Wed Sep 23 Limits and Colimits SA Ch. 5 SA Ch. 5: 1-4, 6
05 Wed Sep 30 Limits and Colimits SA Ch. 5 SA Ch. 5: 7-12
06 Wed Oct 07 Exponentials SA Ch. 6 SA Ch. 6: 2-4, 6, 8-13, 16
XX Wed Oct 14 Fall Break
07 Wed Oct 21 HOL in Set LB-AB Sec. 1-3 Exercises in the section
08 Wed Oct 28 Naturality SA Ch. 7 SA Ch. 7: 4, 6-9, 11-13, 15, 17
09 Wed Nov 04 Categories of Diagrams SA Ch. 8 SA Ch. 8: 1, 3, 6, 7
10 Wed Nov 11 Adjoints SA Ch. 9 SA Ch. 9: 1-5, 8, 9, 11, 17, 18
11 Wed Nov 18 Hyperdoctrines LB-AB Sec. 4 Exercises in the section
12 Wed Nov 25 U-based Hyperdoctrines LB-AB Sec. 5-6.1 Exercises in the section

Where

See below for some hints on the reading.

Handins

Handin Hand out date Hand in deadline Link to PDF
1 Wed Sep 09 Wed Sep 16 Assignment 1
2 Wed Sep 23 Wed Sep 30 Assignment 2
3 Wed Sep 30 Wed Oct 07 Assignment 3
4 Wed Oct 07 Wed Oct 21 Assignment 4
5 Wed Oct 21 Wed Oct 28 Assignment 5
6 Wed Nov 11 Wed Nov 18 Assignment 6
7 Wed Nov 18 Wed Dec 02 Assignment 7

Reading notes

SA Ch. 1

Some of the examples in section 1.4 in the chapter are not relevant for us in the rest of the course. In particular examples 9, 10 and 11.

If you are not familiar with Cayley’s theorem then it is safe to skip that theorem in section 1.5, as well as theorem 1.6.

We will not use free categories in the rest of the course, so this part of section 1.7 is safe to skip. However do read and understand the part about free monoids. They are an example which comes up often.

SA Ch. 2

Projective objects are safe to skip. If you are not already familiar with projective modules, or similar structures, then it is not going to be very meaningful.

The section on generalized elements starts with some very specific examples which, if you are not already familiar with, you do not have to spend time understanding. The important part of that section is on page 36 and onwards.

Examples 5 and 6, and Remark 2.18 in Section 2.5 are safe to skip if you are not familiar with the material. We will see a more systematic presentation of Example 6 later in the course.

SA Ch. 3

In Example 3.6 you can skip the coproduct in Top example if you are not familiar with topological spaces. But do read and understand the coproduct of posets (they also appear later in the chapter).

Example 3.8 is somewhat informal. In particular the notion of equality of proofs. It is safe to skip the details. We will see a more precise treatment of a similar setup later.

Example 3.10, together with Proposition 3.11, is safe to skip.

In Example 3.22 the general setup, with a general notion of an algebra, is perhaps a bit difficult to understand precisely. I suggest you try to understand it in the case of monoids or groups. In particular you should understand the statement and the proof of Proposition 3.24.

SA Ch. 5 (first part)

For the fifth meeting you should read until about Definition 5.15 on page 101. Pullbacks are a very important notion.

SA Ch. 5 (second part)

The example involving Boolean algebras and ultrafilters just after Corollary 5.27 can be skipped. Examples 5.28, 5.29 and 5.32 can be skipped if you are not familiar with the subject matter.

SA Ch. 6

We will go into more details on lambda calculus in the next session, so it is fine to only skim those sections in the chapter. In particular that means Section 6.6 and the part of 6.7 after Definition 6.21.

You can also skip Example 6.6 about exponentials of graphs.

SA Ch. 7

The most important concepts in this chapter are the notions of a functor category, natural transformation and equivalence of categories.

Some of the examples involve notions, such as vector spaces or topological spaces with which you might not be familiar. These can be omitted, but if you are familiar with the concepts then the examples might be useful to read to understand how the abstract definitions generalise the known concepts.

In Example 7.3 you can skip the part about topological spaces and rings on page 151. You can skip section 7.3 on Stone duality. You can skip example 7.12 if you do not know about vector spaces. If you do, then this is a classical example of naturality so useful to know.

You can skip section 7.8 on monoidal categories. They are an interesting and important concept, but to study them in any detail would require a course of its own.

You can skip everything after (and including) Example 7.30 on page 178.

SA Ch. 8

You can skip the part of section 8.1 about simplicial sets.

The most important part of this section is the Yoneda embedding and the Yoneda lemma.

Proposition 8.10 provides a very formal construction which is then used in Propositions 8.12 and 8.13. Although the construction is important and useful to know, it might be difficult to digest. We will cover a more elementary proof, which does not use Proposition 8.10, of the fact that categories of diagrams are cartesian closed (essentially Theorem 8.14) at the meeting. Thus the material in Propositions 8.10, 8.11, 8.12, 8.13 is optional.

You can skip Proposition 8.11.

You can skip the section on topoi (or toposes). We will study closely related notions (hyperdoctrines) later on with more motivation and more concrete examples.

If you are comfortable with Proposition 8.10 then you can try exercises 2 and 8 as well.

SA Ch. 9

You can skip Examples 9.10 and 9.11 and Section 9.5.

One of the most important facts you should remember is that right adjoints preserve limits, and left adjoints preserve colimits.

You can skip everything after Example 9.15 and until (but not including) Section 9.8.

The adjoint functor theorem in general is a non-trivial and subtle result. We will not consider the result in general but do look at Example 9.33 on page 242. It is a special case of the adjoint functor theorem which is much simpler to prove and to use. Thus skip Section 9.8, apart from Example 9.33, until page 246. The final part of Section 9.8 defines and studies the natural numbers object and does not use any of the preceding facts, so please read that.

Exam

At the exam you will randomly pick one of the topics below. Then you can look very briefly at your outline and then you should start presenting something related to the chosen topic, for 13 minutes, and then the examiners will ask you some questions. Time is short so think carefully about what you want to present and how much to write on the board. The exam will last 20 minutes in total.

Exam topics