Welcome to the course page of the Category theory summer project at ILLC


Announcements
 You can find the schedule for the coming lectures and the location below.
 You can post any comments/suggestions regarding the course below.
 If you are interested in this project, please contact Kohei (kishidakohei[at]gmail.com) and Sumit (S.Sourabh[at]uva.nl) by email.

Presentation Schedule
Speaker  Topic  Date/Time/Location 
Facundo Carreiro  Introduction to Coalgebras  13:3014:45, 26th June, G2.13 
Fatemeh Seifan  Relation liftings in Coalgebras  15:0016:30, 26th June, G2.13 

Sumit Sourabh  Introduction to Topos theory  13:3016:30, 28th June, G2.04 

Gianluca Paolini  Monads I  13:3014:20, 2nd July, G2.04 
Hugo Nobrega  Quantified modal logic  14:3015:20, 2nd July, G2.04 

Vlasta Sikimic  Category theoretic proof theory  13:0013:50, 5th July, G2.04 
Femke Bekius  Application of category theory in cognitive science  14:0014:50, 5th July, G2.04 
Giovanni Cina  Category theoretic semantics for FOL  15:0015:50, 5th July, G2.04 
Julia Ilin  Monads II  16:0016:50, 5th July, G2.04 

Lectures
 Lecture 1 (5th June) : Introduction to Category theory; Examples of categories, product category, slice category, opposite category, free monoid construction (Awodey Ch.1)
 Lecture 2 (7th June) : Epis and monos; Initial and terminal objects; Products and coproducts; Equalizers and coequalizers
 Lecture 3 (12th June) : Subobjects; Pullbacks and Pushouts; Inverse image operation; Diagram category; Cones and Cocones
 Lecture 4 (14th June) : Limits and Colimits; Exponentials
 Lecture 5 (19th June) : Full and faithful functors; Natural transformation; Functor category; Equivalence of categories
 Lecture 6 (21st June) : Yoneda lemma and embedding; Adjoint functors; Adjoint fucntor theorem

Homework
The following assignments have to be submitted
 Assignment 1: Choose any 4 exercises out of Ex 4, 5, 7, 8, 13 from Aweodey Ch.1 (the 13th exercise is worth 2 exercises) (Deadline Tuesday 12th June)
 Assignment 2: Choose any 4 out of Ex 2, 3, 8, 14, 15 from Chapter 2 and any 4 out of 3, 4, 11, 13, 15 from Chapter 3 (Deadline Tuesday 12th June)
 Assignment 3: Exercises 2(b), 2(c), 9, 11 of Chapter 2; Proposition 5.5 (pp. 8182) (Deadline Tuesday 19th June)
 Assignment 4: Choose four from Exercises 4, 7, 8, 9, 12 of Chapter 6. (A) The last two paragraphs of Section 6.2 (p. 114):
Prove the equation.Deadline Tuesday 19th June)
 Assignment 5: Choose 4 out of Ex 2, 3, 9, 11, 14, 17 of Chapter 7 (Deadline Tuesday 26th June)
 Assignment 6: Choose 5 out of Ex 7, 8, 11 of Chapter 8 and Ex 1, 2, 6, 19 of Chapter 9 (Deadline Tuesday 26th June)

Discussion problems
 Discussion probelms 1: The following problems will be discussed during the lecture on Thursday 7th June
 Example 8 of Section 1.4 (pp. 910), posets as categories:
Check that the monotone maps between given posets are exactly the
functors between the posets regarded as categories.
 Example 13 of Section 1.4 (pp. 1112), monoids as categories:
Check that the monoid homomorphisms between given monoids are exactly
the functors between the monoids regarded as categories.
 The first sentence of p. 13 on group homomorphisms as functors:
Prove the sentence!
 Read the rest of Section 1.5 (pp. 1314).
 Example 1 of Section 1.6 (pp. 1415), product category:
Check that projection functors are really functors.
 Example 4 of Section 1.6 (pp. 1617), slice category:
Describe how the composition functor g_\ast acts on arrows of
\mathbf{C} / C, and check that it really is a functor. Moreover, check
that the functor \math{C} / () : \mathbf{C} \to \mathbf{Cat} is really
a functor.
 Read the rest of Section 1.7 (pp. 1923) as far as you can.
 Discussion probelms 2: The following problems will be discussed during the lecture on Tuesday 12th June
 Read the rest of Section 2.1 (pp. 2728).
 Read Example 2.14 (p. 34).
 Example 3 of Section 2.5 (p. 38), on product category:
Check the two facts.
 Read Examples 5, 6 of Section 2.5 (pp. 3842).
 Section 2.6 (pp. 4243), on the functor \times : \mathbf{C} \times
\mathbf{C} \to \mathbf{C}:
Check that it really is a functor.
 Definition 3.3 (on p. 49), the last sentence of p. 49:
Show that "injections" (in Sets) are not necessarily injective.
 Read the rest of Section 3.2 (p. 5155) as far as you can.
 Read Example 3.14 (pp. 5657).
 Prove Proposition 3.19 directly, without using (and indeed without
peeking at the proof of) Proposition 3.16 and the duality principle.
 Read the rest of Section 3.4 (p. 6265) as far as you can.
 Discussion probelms 3: The following problems will be discussed during the lecture on Thursday 14th June
 Proposition 5.10 (p. 85):
Check that h^\ast(g /circ f) = h^\ast(g) /circ h^\ast(f).
 Read Examples 5.12 (pp. 8688).
 Examples 5.13 (p. 88):
Work out the details.
 Read the rest of Section 5.6 (pp. 96102).
 Exercise 4 of Chapter 5 (p. 104).
 Discussion problems 4: The following problems will be discussed during the lecture on Thursday 19th June
 Proposition 5.21 (p. 91):
Do a proof by yourself.
 Proposition 5.25 (p. 94):
Do a proof by yourself that the representable functors preserve
equalizers.
 Read Examples 6.5 and 6.6 (pp. 109111).
Also do the exercises given there.
 Read Sections 6.3 and 6.4 (pp. 115120).

General Information
Course Description
This project will introduce students to category theory, a branch of
abstract algebra that has found many applications in mathematics, logic,
and computer science, among others. Category theory provides a framework of
formal methods that enable structural and unificatory approaches to
phenomena that occur commonly across different subjects. The goal of this
project is to acquaint students with these methods and to prepare them to
apply category theory to many topics in logic, computer science and other
fields, such as duality theory, coalgebraic logic, and categorical logic.
The first three weeks of the course will consist of introductory lectures by
the instructor and exercise / discussion sessions, and will cover
fundamental notions and methods in category theory such as adjoints and the
Yoneda lemma. In the final week, students will present materials on applied
topics.
Staff
 Instructor: Kohei Kishida (email: kishidakohei[at]gmail.com)
 Teaching Assistant: Sumit Sourabh (email: S.Sourabh[at]uva.nl)
Dates/location
 Tuesday 5th June, 14:0017:00 hrs in G2.13
 Thursday 7th June, 10:0013:00 hrs in G3.13
 Tuesday 12th June, 13:3016:30 hrs in G2.13
 Thursday 14th June, 10:0013:00 hrs in G3.13
 Tuesday 19th June, 13:3016:30 hrs in G2.13
 Thursday 21st June, 13:3016:30 hrs in G2.04
 Tuesday 26th June, 13:3016:30 hrs in G2.13
 Thursday 28th June, 13:3016:30 hrs in G2.04
 Monday 2nd July, 13:3016:30 hrs in G2.04
 Thursday 5th July, 13:3016:30 hrs in G2.04
Grading
This course is worth 6 ECTS. Students will work on exercise problems and submit homework every week. In
the final week, they will give presentations on topics in logic, computer
science, or other areas that use category theory. Participation in class
discussion will also be assessed.
Prerequisites
Firstorder logic (and familiarity with basic notions in set theory).
Familiarity with basic algebras such as monoids, groups, partial orders and
lattices will be helpful.
References
 The main text for the course will be Category Theory (Oxford Logic Guides 49). Oxford University Press. 2nd edition, 2010 by Steve Awodey.
 Alexander Kurz has complied an excellent list of category theory related books on this page.
 The Catsters channel on Youtube has short videos on category theory related topics.
