A BRIEF SYNOPSIS
(Each class lasted approximately 3 hours, unless specified otherwise)
Class 1 September 23-24, 2026
Organizational details. The subject of category theory. 3 kinds of structures in
mathematics: algebraic, topological, order. Prerequisites: groups, rings,
algebras, vector spaces,
commutative diagrams,
tensor product,
tensor algebra,
free groups.
Polynomial algebras as free objects in the variety of associative commutative
algebras, tensor algebras as free objects in the variety of associative
algebras. Definition of category. Classes vs. sets.
Russell's paradox.
Examples of categories. Small and large categories. Sets as categories with the "minimal possible" number of arrows.
(Slides, pp.3-5,20 and references therein; Pareigis, pp.1-6;
Riehl, pp.x-xi,3-6).
Class 2 September 30, 2026
Monoids as categories with one object and many arrows.
Subcategory, examples. Subcategories of \(\mathbb S_X\) and of \(\mathbb C_M\).
Dual (opposite) categories: definition, examples. Product of categories.
\(\mathbb S_X \times \mathbb S_Y \simeq \mathbb S_{X \times Y}\),
\(\mathbb C_M \times \mathbb C_N \simeq \mathbb C_{M \times N}\).
A bit of history. Definition of a covariant and contravariant functor.
(Slides, pp.6-12 and references therein; Pareigis, pp.7-8; Riehl, pp.9-10,13).
Class 3 October 7, 2026
Discusson of homeworks 1-3. The category \(\mathbf{Cat}\) of small categories.
Composition of functors is a functor. Examples of functors: constant functor,
forgetful functors, \(\mathbf{Group} \to \mathbf{Group}\) sending a group
\(G\) to \(G/Z(G)\), or to \(G/[G,G]\).
There is no functor \(\mathbf{Group} \to \mathbf{Group}\) sending a group \(G\) to \(Z(G\)).
(Slides, pp.13,21 and references therein; Pareigis, p.8; Riehl, pp.13-18).
Class 4 October 14, 2026
Discussion of homeworks 4-7. Further examples for functors. Isomorphism of
categories, examples. \(\mathbb S_X \simeq \mathbb S_Y\) iff \(|X| = |Y|\).
\(\mathbb C_M \simeq \mathbb C_N\) iff \(M \simeq N\).
Isomorphism of objects within a category.
Skeleton of a category.
Equivalence of categories. Categories of matrices and finite-dimensional vector
spaces are not isomorphic, but equivalent.
(Slides, pp.15-18 and references therein; Adamek et al., pp.34,36).
Class 5 October 21, 2026
An isomorphism of categories is an equivalence. Functors preserve isomorphism
inside categories. Equivalence of categories is an equivalence relation (after
Adamek et al., p.37). Natural transformations. The category of all functors
between two small categories. Examples of natural transformations: determinant
as a natural transformation between functors \(GL_n\) and \(()^*\), group
abelianization.
(Slides, pp.25-26 and references therein; Adamek et al., pp.34-38).
(referred below as "Slides")
