Class 1 September 23, 2025
Organizational details. The subject of category theory. 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.
Small and large categories. Subcategory, examples.
Dual (opposite) categories: definition, examples. Product of categories.
(Slides, pp.3-8,20 and references therein; Pareigis, pp.1-6;
Riehl, pp.3-6,9-10).
Class 2 September 30, 2025
\(\mathbb S_X \times \mathbb S_Y \simeq \mathbb S_{X \times Y}\),
\(\mathbb C_M \times \mathbb C_N \simeq \mathbb C_{M \times N}\).
A bit of history. Definition of a covariant and contravariant functor, examples.
Recap of
tensor products.
Examples of functors \(\mathbf{Group} \to \mathbf{Group}\) sending
a group \(G\) to \(G/Z(G)\), or to \(G/[G,G]\).
Forgetful functors. Composition of functors is a functor.
Isomorphism of categories.
\(\mathbb S_X \simeq \mathbb S_Y\) iff \(|X| = |Y|\).
\(\mathbb C_M \simeq \mathbb C_N\) iff \(M \simeq N\).
The category \(\mathbf{Cat}\) of small categories.
(Slides, pp.9-13,15,21 and references therein; Pareigis, pp.6-8;
Riehl, pp.13-18).
Class 3 October 7, 2025
Discussion of homeworks 1-3. There is no functor \(\mathbf{Group} \to \mathbf{Group}\) sending a group \(G\) to \(Z(G\)).
Isomorphism of objects within a category. Equivalence of 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.
(Slides, pp.16-18 and references therein; Adamek et al., pp.34-38).
Class 4 October 14, 2025
Discussion of homeworks 4-6. Categories of matrices and finite-dimensional
vector spaces are equivalent.
Skeleton of a category.
(Slides, p.18 and references therein, Adamek et al., p.36).
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 algebra,
free group.
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.
(referred below as "Slides")
