This is a webpage for the course “Introduction to Categories” at ENS Paris-Saclay in the academic year 2026-2027.

Below you can find some useful information for the course.

Schedule

The course starts on Monday September 7 and runs until Wednesday October 21, with an exam on Wednesday November 4.

Lectures

We have covered aspects from the following topics in the course so far:

  1. Monday 07 September: definition of category, monomorphism, epimorphism, isomorphism, locally small categories; opposite category, functors; basic propositions related to these notions; examples in Set, Vect, Pos, Cat, and other categories.
  2. Monday 14 September: commutative diagrams, terminal object, binary products, finite products, basic properties and propositions related to (finite) products, products are unique up to coherent isomorphism, binary products as a functor; examples in Set and Vect.
  3. Monday 21 September: Limits and colimits, terminal object as a limit of the empty diagram, binary/small products as a limit, (co)limits are unique up to coherent isomorphism, (co)equalisers, example: equalisers in Set, preservation of (co)limits by a functor.
  4. Monday 28 September: Solved exercises 21 and 22, a category C with binary products has a functor (- x -) : C x C –> C, natural transformations, example: identity natural transformation, example: natural transformations Id ==> P and P^2 ==> P, where P : Set –> Set is the powerset functor, example: in a category C with binary coproducts one can define a natrual transformation with components beta_A : A + A –> A, natural transformations compose, def: functor category.
  5. Monday 5 October: universal arrows (for adjunctions), adjunctions (definition via natural bijections), universal arrows determine adjunctions, mentioned that adjunctions also determine (co)universal arrows but without much details, example: adjunction between Set and Vect, example: adjunction between Set and Pos.

Exercises

Exercises, to be discussed on Wednesdays, are available here.

Homework is assigned at the end of each exercise class, to be be handed in the following Wednesday by 10:45, either via e-mail [email protected], or in person.

  1. 9 September: exercises 1-5 (except 5e). Homework for 16 Sep: exercise 7.

  2. 16 September: exercises 6, 8, 10, 11, 13, 14, 15a. Homework for 23 Sep: exercise 12.

  3. 23 September: exercise 15, 18, 19, 20ab. (21 and 22 are treated in 28 September’s lecture.) Homework for 30 Sep: exercise 24.

  4. 30 September: exercise 27, 28, 29. Homework for 7 Oct: exercise 30a.

Further Reading

If you wish to learn more about category theory, we recommend the following texts:

About Yoneda’s Lemma (as mentioned in TD):

Evaluation

Teachers