6/7TEKA Category Theory, winter semester 2025/2026
Class 3 October 7, 2025
Equivalence of categories is an equivalence relation (after
Adamek et al., p.37).
(Slides, pp.16-18 and references therein; Adamek et al., pp.34-38).
Class 4 October 14, 2025
Self-study assignment: natural transformations. Put
attention to examples (determinant, group multiplication, group abelianization,
etc.)
(Slides, pp.25-26 and references therein).
Class 5 October 21, 2025
Natural transformations.
Self-study assignment for the next few weeks:
Yoneda lemma.
Adjoint functors.
Free objects as adjoint
functors to forgetful functors.
(Slides, pp.35-42 and references therein; Mac Lane, pp.79-81,86-87;
Riehl, pp.116-121).
Class 6 November 11, 2025
Yoneda lemma, adjoint functors, free objects.
Self-study assignment for the next week:
Limits and
colimits (definition, examples:
p-adic numbers, formal
power series, direct products and direct sums). Initial and terminal objects.
(Slides, pp.30-33 and references therein).
Class 7 November 18, 2025
Discussion of homework 13. Limits and colimits, initial and terminal objects.
Class 8 December 2, 2025
Discussion of homeworks 15-16. Examples of existence and non-existence of limits
and colimits.
Self-study assignment for the next week: Preservation of limits. Continuous
functors. Adjoints on limits.
(Mac Lane, pp.116-120; Bergman, pp.342-353; Adamek et al., pp.223-227).
Class 9 December 16, 2025
Preservation of limits.
HOMEWORKS
6/7TEKA Category Theory, winter semester 2024/2025
A BRIEF SYNOPSIS
Class 4 October 15, 2024
There is no "natural" (and also "not so natural") functor from \(\mathbf{Set}\)
to \(\mathbf{Group}\) sending a set \(X\) to the group of its bijections
\(S(X)\).
Class 5 October 22, 2024
Equivalence between categories of type \(\mathbb S_X\) and \(\mathbb C_M\).
(Slides, pp.16-18 and references therein; Adamek et al., pp.34-38).
Class 6 November 5, 2024
Discussion of homeworks 4-10. Natural transformations. Determinant as a
natural transformation between functors \(GL_n\) and \(()^*\).
(Slides, pp.25-26 and references therein).
Class 7 November 19, 2024
Discussion of homework 11. Further examples of natural transformations
(group abelianization, group multiplication). The category of all functors
between two small categories. The Yoneda lemma and Yoneda embedding. Cayley
theorem about groups as a corollary of the Yoneda lemma.
(Slides, pp.26,35-38 and references therein; proof of Yoneda lemma according to
Wikipedia).
Class 8 November 28, 2024
Discussion of homeworks 12-13.
Universal property of free objects. Adjoint functors (definition, examples).
(Slides, pp.40-41 and references therein; Mac Lane, pp.79-81,86-87;
Riehl, pp.116-121).
HOMEWORKS
6TEKA Category Theory, winter semester 2023/2024
A BRIEF SYNOPSIS
Class 1 November 14, 2023 (approx. 3h 10min)
Prerequisites: semigroups, monoids,
lattices,
homomorphism and isomorphism of algebraic structures.
Class 3 November 28, 2023 (approx. 3h 30min)
\(\mathbb C_M\) is equivalent
to \(\mathbb C_N\) iff \(\mathbb C_M\) is isomorphic to \(\mathbb C_N\) iff
\(M \simeq N\).
(Slides, pp.15-18,21 and references therein).
Class 4 December 5, 2023 (approx. 3h 10min)
Discussion of homework 2. \(\mathbb S_X\) is equivalent to \(\mathbb S_Y\) iff
\(\mathbb S_X\) is isomorphic to \(\mathbb S_Y\) iff \(|X| = |Y|\). Functors
preserve isomorphism inside categories. Equivalence of categories is an
equivalence relation (after Adamek et al., p.37). Natural
transformations. Examples: determinant, group abelianization.
(Slides, pp.18,25-26 and references therein; Adámek et al., pp.34-38).
Class 5 December 12, 2023 (approx. 3h 10min)
Discussion of homeworks 3-13. The Yoneda lemma and its corollaries. Cayley
theorem about groups as a corollary of the Yoneda lemma.
(Slides, pp.35-38 and references therein; proof of Yoneda lemma according to
Wikipedia).
HOMEWORKS
6TEKA Category Theory, winter semester 2022/2023
A BRIEF SYNOPSIS
Each class lasted approximately 3 hours, unless specified otherwise.
Class 1 September 20, 2022
Prerequisites: semigroups, monoids,
lattices,
homomorphism and isomorphism of algebraic structures.
Class 3 October 4, 2022
A category is small iff the class of its arrows is a set.
(Slides, pp.8,11-13 and references therein; Pareigis, p.8; Riehl, pp.13-18).
Class 4 October 11, 2022
Discussion of homeworks 5,6. No "natural" functor
\(\mathbf{Set} \to \mathbf{Grp}\) sending \(X\) to \(S(X)\).
(Slides, pp.15,21 and references therein).
Class 5 November 1, 2022 (approx. 3h20min)
An isomorphism of categories is an
equivalence. Equivalence of categories is an equivalence relation (after Adamek
et al., p.37). Examples of equivalence of categories with finite number of
objects and arrows. \(\mathbb C_M\) is equivalent to \(\mathbb C_N\) iff
\(M \simeq N\).
(Slides, pp.15-18,20 and references therein, Adamek et al., pp.34-38).
Class 6 November 8, 2022 (approx. 3h40min)
Discussion of homeworks 7-14. Natural transformations. Examples: determinant,
group abelianization, and group multiplication as natural transformations.
The category of all functors between two small categories.
(Slides, pp.25-26,37 and references therein).
Class 7 November 15, 2022 (approx. 2h50min)
Discussion of homework 15. The Yoneda lemma and its corollaries. Natural
transformations between functors \(\mathbf C_G \to \mathbf{Set}\) correspond to
\(G\)-equivariant maps. Cayley theorem about groups as a corollary of the Yoneda
lemma.
(Slides, pp.35-36,38 and references therein; proof of Yoneda lemma
according to
Wikipedia).
Class 8 November 22, 2022 (approx. 3h20min)
Discussion of homework 16. Adjoint functors (definition, examples).
(Slides, pp.40-41 and references therein; Mac Lane, pp.79-81,86-87;
Riehl, pp.116-121).
Class 9 November 29, 2022 (approx. 3h10min)
Discussion of homework 17. Category of arrows of a given category. Kernels as
adjoint functor (redo). Natural equivalence as adjoint functors. Composition of
adjoint functors is adjoint functor. Adjoint functors are unique up to natural
equivalence.
(Slides, p.42).
Class 10 December 6, 2022 (approx. 3h10min)
Discussion of homeworks 18-19. Category attached to a poset.
Limits and colimits
(definition, examples). Initial and terminal objects.
(Slides, pp.30-33 and references therein).
Class 11 December 3, 2022 (approx. 2h20min)
Discussion of homeworks 20-21. Free objects in an arbitrary concrete category.
(Bergman, p.264).
HOMEWORKS
6/7TEKA Category Theory, winter semester 2021/2022
Material not formally covered in the last course:
Refresher: groups given by generators and relations,
universal properties.
The diehedral group \(D_3\)
given by generators and relations.
Equivalence of categories of the form \(\mathbb C_M\) and of the form \(\mathbb S_X\).
HOMEWORKS
6TEKA/7TEKA/TEKAT Category Theory, winter semester 2020/2021
A BRIEF SYNOPSIS
Each class lasted approximately 1h30min, unless specified otherwise.
(videos: to watch on youtube, send me a request for access)
Class 1 September 24, 2020 (approx. 45 min)
The subject of category theory. Organizational details.
youtube video
Class 2 September 25, 2020 (approx. 1h50 min)
Refresher: vector spaces, groups,
semigroups, rings,
algebras, lattices,
various types of algebraic structures.
Tensor product.
youtube video (partially)
Class 3 October 2, 2020
Refresher: tensor product (continuation),
free group,
free objects in algebraic systems,
universal properties, isomorphism,
commutative diagrams.
Class 4 October 8, 2020
Refresher: polynomial algebras as free objects in the variety of associative
commutative algebras, tensor algebras as free objects in the variety of
associative algebras.
youtube video
Class 5 October 9, 2020
Definition of category.
Classes vs sets.
Examples of categories. Categories with a small number of objects.
youtube video
Class 6 October 15, 2020
Sets as categories with the "minimal possible" number of arrows. Groups and
monoids as categories with one objects and many arrows. Dual (opposite)
categories: definition, examples.
youtube video
Class 7 October 16, 2020
Subcategories, examples. Subcategories of the category attached to a monoid.
youtube video (partially)
Class 8 October 22, 2020 (approx. 1h50min)
Discussion of Homeworks 1 and 2. Product of categories, examples. The category
attached to the direct product of monoids \(G \times H\) is isomorphic to the
product of categories attached to \(G\) and \(H\).
History (Emmy Noether, Eilenberg, Mac Lane).
youtube video (partially)
Class 9 October 23, 2020
Covariant and contravariant functors, examples.
youtube video
Class 10 October 29, 2020
Further examples of functors (quotient by the center, quotient by the commutator
of a group) and non-functors
(center of a group).
youtube video
Class 11 November 5, 2020 (approx. 2 hours)
Notion of an isomorphism (arrow) in a category. Isomorphism of
categories, examples. Equivalence of categories. Example: the "category of matrices"
\(\mathbf{Mat}\) is equivalent, but not isomorphic to the category of
finite-dimensional vector spaces \(\mathbf{FVect}\).
youtube video
Class 12 November 6, 2020 (approx. 1h40min)
Theorem: a functor maps isomorphic objects to isomorphic objects, and an
isomorphism (in a source category) to an isomorphism (in a target category).
Composition of maps between categories. Theorem: composition of functors is a
functor. Equivalence of categories is an equivalence relation (according to
Adamek et al., p.37).
youtube video
Class 13 November 12, 2020 (approx. 1h10min)
Small and large categories. A category is small if and only if the class of its
arrows is a set. The category \(\mathbf{Cat}\).
youtube video
Class 14 November 13, 2020 (approx. 2h)
Concrete categories, concrete functors. Natural transformations.
youtube video
December 2, 2020: Self-study assignment: Universal arrow,
universal element, limit, colimit (pp. 28-32 of
the slides). Follow references to the literature at the slides, and put special attention to
concrete instances of these things, like
p-adic numbers, formal power series
(p.30), direct products and direct sums (p.32).
Class 15 December 7, 2020 (approx. 2h30min)
Examples of direct and inverse limits.
Proof of theorem from p. 22 of the slides to the effect that any small category
is concrete (according to
wikipedia).
Adjoint functors, examples. The composition of adjoint functors is adjoint.
Class 16 December 9, 2020 (approx. 2h30min)
Yoneda lemma, its consequence (Yoneda embeding)
(the proof of Yoneda lemma is according to
wikipedia).
Discussion of homeworks 1-9.
Class 17 December 11, 2020 (approx. 2h)
Discussion of homeworks 10-15.
HOMEWORKS
Created: Mon Aug 19 2019
Last modified: Wed Oct 14 2026 19:01:03 CEST