Category theory

June-July 2012

Institute for Logic, Language and Computation

Universiteit van Amsterdam

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


  • 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] and Sumit (S.Sourabh[at] by email.

Presentation Schedule

Facundo Carreiro Introduction to Coalgebras 13:30-14:45, 26th June, G2.13
Fatemeh Seifan Relation liftings in Coalgebras 15:00-16:30, 26th June, G2.13

Sumit Sourabh Introduction to Topos theory 13:30-16:30, 28th June, G2.04

Gianluca Paolini Monads I 13:30-14:20, 2nd July, G2.04
Hugo Nobrega Quantified modal logic 14:30-15:20, 2nd July, G2.04

Vlasta Sikimic Category theoretic proof theory 13:00-13:50, 5th July, G2.04
Femke Bekius Application of category theory in cognitive science 14:00-14:50, 5th July, G2.04
Giovanni Cina Category theoretic semantics for FOL 15:00-15:50, 5th July, G2.04
Julia Ilin Monads II 16:00-16:50, 5th July, G2.04


  • 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


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. 81--82) (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
    1. Example 8 of Section 1.4 (pp. 9--10), posets as categories: Check that the monotone maps between given posets are exactly the functors between the posets regarded as categories.
    2. Example 13 of Section 1.4 (pp. 11--12), monoids as categories: Check that the monoid homomorphisms between given monoids are exactly the functors between the monoids regarded as categories.
    3. The first sentence of p. 13 on group homomorphisms as functors: Prove the sentence!
    4. Read the rest of Section 1.5 (pp. 13--14).
    5. Example 1 of Section 1.6 (pp. 14--15), product category: Check that projection functors are really functors.
    6. Example 4 of Section 1.6 (pp. 16--17), 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.
    7. Read the rest of Section 1.7 (pp. 19--23) as far as you can.
  • Discussion probelms 2: The following problems will be discussed during the lecture on Tuesday 12th June
    1. Read the rest of Section 2.1 (pp. 27--28).
    2. Read Example 2.14 (p. 34).
    3. Example 3 of Section 2.5 (p. 38), on product category: Check the two facts.
    4. Read Examples 5, 6 of Section 2.5 (pp. 38--42).
    5. Section 2.6 (pp. 42--43), on the functor \times : \mathbf{C} \times \mathbf{C} \to \mathbf{C}: Check that it really is a functor.
    6. Definition 3.3 (on p. 49), the last sentence of p. 49: Show that "injections" (in Sets) are not necessarily injective.
    7. Read the rest of Section 3.2 (p. 51--55) as far as you can.
    8. Read Example 3.14 (pp. 56--57).
    9. Prove Proposition 3.19 directly, without using (and indeed without peeking at the proof of) Proposition 3.16 and the duality principle.
    10. Read the rest of Section 3.4 (p. 62--65) as far as you can.
  • Discussion probelms 3: The following problems will be discussed during the lecture on Thursday 14th June
    1. Proposition 5.10 (p. 85): Check that h^\ast(g /circ f) = h^\ast(g) /circ h^\ast(f).
    2. Read Examples 5.12 (pp. 86--88).
    3. Examples 5.13 (p. 88): Work out the details.
    4. Read the rest of Section 5.6 (pp. 96--102).
    5. Exercise 4 of Chapter 5 (p. 104).
  • Discussion problems 4: The following problems will be discussed during the lecture on Thursday 19th June
    1. Proposition 5.21 (p. 91): Do a proof by yourself.
    2. Proposition 5.25 (p. 94): Do a proof by yourself that the representable functors preserve equalizers.
    3. Read Examples 6.5 and 6.6 (pp. 109--111). Also do the exercises given there.
    4. Read Sections 6.3 and 6.4 (pp. 115--120).

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.


  • Instructor: Kohei Kishida (email: kishidakohei[at]
  • Teaching Assistant: Sumit Sourabh (email: S.Sourabh[at]


  • Tuesday   5th June,   14:00-17:00 hrs in G2.13
  • Thursday 7th June10:00-13:00 hrs in G3.13
  • Tuesday   12th June, 13:30-16:30 hrs in G2.13
  • Thursday 14th June, 10:00-13:00 hrs in G3.13
  • Tuesday   19th June, 13:30-16:30 hrs in G2.13
  • Thursday 21st June, 13:30-16:30 hrs in G2.04
  • Tuesday   26th June, 13:30-16:30 hrs in G2.13
  • Thursday 28th June, 13:30-16:30 hrs in G2.04
  • Monday     2nd July, 13:30-16:30 hrs in G2.04
  • Thursday 5th July13:30-16:30 hrs in G2.04


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.


First-order logic (and familiarity with basic notions in set theory). Familiarity with basic algebras such as monoids, groups, partial orders and lattices will be helpful.


  • 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.