6TEKA Category Theory, winter semester 2023/2024
A BRIEF SYNOPSIS
Class 1 November 14, 2023 (approx. 3h 10min)
Organizational details. The subject of category theory. 3 kinds of structures in
mathematics: algebraic, topological, order. Prerequisites: groups, semigroups,
monoids, rings, algebras, vector spaces,
tensor product,
free groups,
commutative diagrams,
lattices,
homomorphism and isomorphism of algebraic structures. Definition of category.
Classes vs. sets.
Russell's paradox.
Examples of categories. Monoids as categories with one object and many arrows.
Sets as categories with the "minimal possible" number of arrows.
(Slides, pp.3-5 and references therein; Pareigis, pp.1-6; Riehl, pp.x-xi,3-6).
Class 2 November 21, 2023 (approx. 3h 30min)
Small and large categories. Dual (opposite) categories: definition, examples.
Subcategories, examples. Subcategories of \(\mathbb S_X\) and of
\(\mathbb C_M\). Product of categories.
\(\mathbb S_X \times \mathbb S_Y \simeq \mathbb S_{X \times Y}\),
\(\mathbb C_M \times \mathbb C_N \simeq \mathbb C_{M \times N}\).
Covariant and contravariant functors. Forgetful functors. Examples.
A bit of history.
(Slides, pp.6-13 and references therein; Pareigis, p.8; Riehl, pp.9-10,13-18).
Class 3 November 28, 2023 (approx. 3h 30min)
Discussion of homework 1. Further examples of functors
\(\mathbf{Groups} \to \mathbf{Groups}\):
\(G \to G/Z(G), G \to G/[G,G], G \to [G,G]\). Example of non-functor:
\(G \to Z(G)\). Composition of functors is a functor. Constant functor (sending
all object to a fixed object, and all arrows to the identity around this
object). The category \(\mathbf{Cat}\) of small categories. Isomorphism of
categories, examples. Equivalence of categories. \(\mathbb C_M\) 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). Categories of matrices and
finite-dimensional vector spaces are not isomorphic, but equivalent. 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
Organizational details. The subject of category theory. 3 kinds of structures in
mathematics: algebraic, topological, order. Prerequisites: groups, semigroups,
monoids, rings, algebras, vector spaces,
tensor product,
free groups,
commutative diagrams,
lattices,
homomorphism and isomorphism of algebraic structures. Definition of category.
Classes vs. sets.
Russell's paradox.
Examples of categories. Sets as categories with the "minimal possible" number
of arrows.
(Slides, pp.3-5 and references therein; Pareigis, pp.1-6; Riehl, pp.x-xi,3-6).
Class 2 September 27, 2022
Monoids as categories with one object and many arrows. Dual (opposite)
categories: definition, examples. Subcategories, examples. Subcategories of
\(\mathbb S_X\) and of \(\mathbb C_M\). Product of categories.
\(\mathbb S_X \times \mathbb S_Y \simeq \mathbb S_{X \times Y}\),
\(\mathbb C_M \times \mathbb C_N \simeq \mathbb C_{M \times N}\). Definition of
a small category.
(Slides, pp.6-8 and references therein; Riehl, pp.9-10).
Class 3 October 4, 2022
Discussion of homeworks 1-4. A category is small iff the class of its arrows is
a set. Covariant and contravariant functors. Forgetful functors. Examples.
(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 \(S(X)\). Further examples of
functors \(\mathbf{Groups} \to \mathbf{Groups}\):
\(G \to G/Z(G), G \to G/[G,G], G \to [G,G]\). Example of non-functor:
\(G \to Z(G)\). Composition of functors is a functor. Constant functor. The
category \(\mathbf{Cat}\) of small categories. Isomorphism of categories,
examples. (Slides, pp.15,21 and references therein).
Class 5 November 1, 2022 (approx. 3h20min)
Equivalence of categories. Categories of matrices and finite-dimensional vector
spaces are not isomorphic, but equivalent. 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.
Polynomial algebras as free objects in the variety of associative commutative
algebras, tensor algebras as free objects in the variety of associative
algebras.
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: Tue Nov 5 2024 07:03:32 CET