Category Theory, Spring 2008

This is an introductory course in category theory. We follow Steve Awodey's book for most of the course, but also include a few research papers applying category theory in computer science.

The course consists of two hours weekly lectures, followed by a 1 hour optional class used for exercises and discussions. To pass the course, each student must have half the weekly mandatory written exercises accepted and must take responsibility for 1 hour of lectures.

News

At the lecture May 15 we will skip pages 199 - 209
The course has been moved to Thursdays for the remaining few weeks, with the exception of Friday May 9. The new meeting times and rooms are:
- Friday May 9, 13-16 in room 2A12
- Thursdays May 15, 22, 29, 9-12 in room 2A12
Hugo found some interesting lectures on category theory on youtube given by Eugenia Cheng and Simon Willerton, see for example this one. These are perhaps mainly for the more mathematically minded among you.
On Friday March 28 we plan to cover p 125-126, and sections 7.4 - 7.7 skipping example 7.12.
On Friday March 14 we skipped section 6.3 in the lecture.
Friday Feb 29 we will meet in 2A20 from 10 to 12. After lunch we have 2A14.

Time and place

Fridays 10 - 12 and 13 - 14 in 2A.12 at ITU. (Notice that by 10, we mean 10:00 (as opposed to 10:15)). The first week we only meet 10 - 12. The course will last 14 weeks.

Teachers

Thomas Hildebrandt and Rasmus E. Møgelberg

Material

[SA] Steve Awodey: Category theory. Oxford University Press.
[GW] Glynn Winskel: Relations in concurrency. Invited talk at LICS 2005
[EM] Eugenio Moggi: Computational lambda-calculus and monads. LICS 1989.

Suggested additional material

Saunders MacLane: Categories for the working mathematician
Marcelo Fiore: Rough Notes on Presheaves

Tentative Schedule:

Date Speaker Reading Mandatory exercises (hand in week after) Suggestions for ex. class

Fri Feb 8 REM [SA] Chapter 1 [SA] 1.9.4. Show that the inverse of an isomorphism is unique.

Fri Feb 15 Espen, REM [SA] Chapter 2 [SA] 2.9.2, 2.9.7

Fri Feb 22 Morten, REM [SA] Chapter 3 Using the universal mapping property, show that the free monoid construction of [SA] Section 1.7 defines a functor from the category of sets to the category of monoids. [SA] 3.5.4 [SA] 2.9.3, 2.9.6, but also this one

Fri Feb 29 Kasper, REM [SA] Chapter 5 - 5.3 (including) [SA] 5.7.2 and this one [SA] 3.5.1, 3.5.5

Fri Mar 7 REM [SA] Rest of chapter 5 [SA] 5.7.5 and this one [SA] 5.7.1, prove Lemma 5.10

Fri Mar 14 Jesper, REM [SA] Chapter 6 [SA] 6.6.4 (email Rasmus if you need a hint for the last part) and prove last equation p. 112 [SA] 5.7.4 and this one

Fri Mar 28 Hugo, REM [SA] Chapter 7 (first half) [SA] 7.10.5 and 7.10.6 [SA] 6.6.1 and this one

Fri Apr 4 REM, TH [SA] Sections 7.8 and 7.9 and start of Chapter 8 Mandatory bonus Cancelled

Fri Apr 11 TH [SA] Chapter 8 [SA] Ex 8.9.1 and 8.9.2 Bonus exercise

Fri Apr 25 TH [GW] Ex 8 [SA] Ex 8.9.4

Fri May 9 REM, Tom [SA] Chapter 9 Finish example 9.3 in [SA], Exercise 9.9.4

Thurs May 15 REM [SA] Chapter 9 Exercise 9 Bonus exercise again

Thurs May 22 REM, Taus [SA] Chapter 10 [SA] Ex 10.6.4 and show that the Eilenberg Moore category for the free monoid functor on the category of sets is equivalent to the category of monoids. [SA] Ex 9.9.9 and 10.6.9

Thurs May 29 REM [EM] Construct a strength for the side-effect monad ([EM] Ex 1.3) and verify the axioms. [SA] 10.6.8 and 10.6.3

Date			Speaker	Reading	Mandatory exercises (hand in week after)	Suggestions for ex. class
Fri	Feb	8	REM	[SA] Chapter 1	[SA] 1.9.4. Show that the inverse of an isomorphism is unique.
Fri	Feb	15	Espen, REM	[SA] Chapter 2	[SA] 2.9.2, 2.9.7
Fri	Feb	22	Morten, REM	[SA] Chapter 3	Using the universal mapping property, show that the free monoid construction of [SA] Section 1.7 defines a functor from the category of sets to the category of monoids. [SA] 3.5.4	[SA] 2.9.3, 2.9.6, but also this one
Fri	Feb	29	Kasper, REM	[SA] Chapter 5 - 5.3 (including)	[SA] 5.7.2 and this one	[SA] 3.5.1, 3.5.5
Fri	Mar	7	REM	[SA] Rest of chapter 5	[SA] 5.7.5 and this one	[SA] 5.7.1, prove Lemma 5.10
Fri	Mar	14	Jesper, REM	[SA] Chapter 6	[SA] 6.6.4 (email Rasmus if you need a hint for the last part) and prove last equation p. 112	[SA] 5.7.4 and this one
Fri	Mar	28	Hugo, REM	[SA] Chapter 7 (first half)	[SA] 7.10.5 and 7.10.6	[SA] 6.6.1 and this one
Fri	Apr	4	REM, TH	[SA] Sections 7.8 and 7.9 and start of Chapter 8	Mandatory bonus	Cancelled
Fri	Apr	11	TH	[SA] Chapter 8	[SA] Ex 8.9.1 and 8.9.2	Bonus exercise
Fri	Apr	25	TH	[GW]	Ex 8	[SA] Ex 8.9.4
Fri	May	9	REM, Tom	[SA] Chapter 9	Finish example 9.3 in [SA], Exercise 9.9.4
Thurs	May	15	REM	[SA] Chapter 9	Exercise 9	Bonus exercise again
Thurs	May	22	REM, Taus	[SA] Chapter 10	[SA] Ex 10.6.4 and show that the Eilenberg Moore category for the free monoid functor on the category of sets is equivalent to the category of monoids.	[SA] Ex 9.9.9 and 10.6.9
Thurs	May	29	REM	[EM]	Construct a strength for the side-effect monad ([EM] Ex 1.3) and verify the axioms.	[SA] 10.6.8 and 10.6.3