Category Theory, Fall 2020
Meeting schedule
We meet every Wednesday from 10:0011:00 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: 15, 1315, 17, 18 
03  Wed Sep 16  Duality  SA Ch. 3  SA Ch. 3: 14, 6(monoids), 1014 
04  Wed Sep 23  Limits and Colimits  SA Ch. 5  SA Ch. 5: 14, 6 
05  Wed Sep 30  Limits and Colimits  SA Ch. 5  SA Ch. 5: 712 
06  Wed Oct 07  Exponentials  SA Ch. 6  SA Ch. 6: 24, 6, 813, 16 
XX  Wed Oct 14  Fall Break  
07  Wed Oct 21  HOL in Set  LBAB Sec. 13  Exercises in the section 
08  Wed Oct 28  Naturality  SA Ch. 7  SA Ch. 7: 4, 69, 1113, 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: 15, 8, 9, 11, 17, 18 
11  Wed Nov 18  Hyperdoctrines  LBAB Sec. 4  Exercises in the section 
12  Wed Nov 25  Ubased Hyperdoctrines  LBAB Sec. 56.1  Exercises in the section 
Where

SA refers to the book Category Theory by Steve Awodey. Second edition.

LBAB refers to our lecture notes on categorical logic. They are available online.
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 
Exam
The exam will take place over Zoom on January 7th, 2021. The ordering of participants will be announced over email soon.
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
 Limits and colimits
 Exponentials
 Naturality
 Categories of Diagrams
 Adjoints
 Hyperdoctrines
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 Cayleyâ€™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 nontrivial 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.