# Kategoriteori - vt14

Category theory, 7.5 hp, VT2014. Wednesdays 1515-1700 (start 22/1), room 32, house 5.

Description: Category theory, as a tool of describing and reasoning about various structural concepts in a uniform way, is by now a common object in the toolboxes of mathematicians as well as of theoretical computer scientists, and with increasing applications also in other fields. The course intends to give a systematic first introduction to the concepts and methods of category theory to the interested graduate student  and to prepare the student for the application and further development of categorical methods in the student's own particular field.

Prerequisites: A certain level of mathematical maturity will be assumed. (Kunskaper motsvarande kandidatexamen i ett matematikbesläktat ämne. Minst 30 hp matematik på avancerad nivå. Engelska B eller motsvarande)
Textbook: Awodey: "Category theory"
Mac Lane: "Cateories for the working matematician"
Mac Lane and Moerdijk: "Sheaves in geometry and logic",
Johnstone: "Sketches of an elephant, Vol I".
Examination: Written exam. (PhD students are in addition required to give an in-class presentation of an approved topic of their own choosing)

This a PhD level course also suitable for Master students. Master students can take the course under the heading "Selected Topics in Mathematics - Mathematical Logic".
• ### Dag 1

Start up and introduction to the course.

Read: Awodey Introduction & Ch 1. (Mac Lane to 1.4)

Do: Exc. 1,2,3,5,6,7,8, anything else that looks fun, and 14 if you have Mac Lane.

• ### Dag 2

Remember room change! Room 32, house 5.

Free monoids, free categories, products, epis, monos, initial and terminal objects.

Read: Awodey 1.7 and ch 2.

Do: 1.11b, 2.1, 2.3, 2.4.

Read section 2.1.1 (which I forgot to cover), then do 2.5-10.

2.11

Prove Proposition 2.17 and do exercise 2.13.

• ### Dag 3

Representable functors and presheaves. We have a first look at presheaf categories and rehearse some of the notions weve seen so far.

Awodey chapter 3, in so far as there is time for it.

Do: Prove that the Yoneda embedding preserves products. Prove that it sends monomorphisms to pointwise 1-1 natural transformations. Do any exercises from the book that look useful. (Exercises from the book are likely to show up on the exam.)

• ### Dag 4

Finite limits: equalizers and pullbacks. Coproducts and co equalizers. Subobjects. If time well start with ch 4.

Read: Awodey ch. 3 and 5.1-5.3. Especially 3.22, 5.1, and 5.9-5.13 which we did not have time to cover properly.

Do: Describe coproducts in Ab. Describe coequalizers in Mon. Prove the two pullback lemma. 3.1-4, 3.7-10, 5.1-6.

Note: You should at least read the exercises, there are things in there that are quite useful.

• ### Dag 5

Internal groups.

Limits and colimits.

Do: Exercises! From ch 4 and 5, your choice. Feel free to bring them up in class.

• ### Dag 6

Naturality

• ### Dag 7

Naturality. Yoneda Lemma.

We finish chapter 7 and start on chapter 8. We will do chapters 8 and 9 more carefully and in more detail than the previous chapters.

Do: Read chapter 7 and 8.1-8.3. Try proving the Yoneda Lemma from where we left off in class.

• ### Dag 8

Yoneda Lemma and applications. Awodey chapter 8.

• ### Dag 9

We continue Awodey chapter 8 and 9.

I will make a set of hand-in exercises covering the material we have seen so far and post it here. This is so you can make sure that you are on top of the material covered, and to prepare you for the exam.

• ### Dag 10

Elementary toposes.

NB! The midterm exercise set was posted Friday 3/28 (top of the page). It is due the first class after Easter.

• ### Dag 11

Kan extensions

Natural Numbers objects

• ### Dag 12

Student lecture:

Felix Wierstra

Homotopy and Model Categories

Abstract: During this lecture I will explain the basics of homotopy theory and show how this can be formalized by defining model categories. After defining model categories I might give a few applications.