Class 1 September 24, 2024
Organizational details. The subject of category theory. 3 kinds of structures in
mathematics: algebraic, topological, order.
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. Subcategories of
\(\mathbb S_X\) and of \(\mathbb C_M\).
(Slides, pp.3-5,7 and references therein; Pareigis, pp.1-6; Riehl, pp.3-6).
Class 2 October 1, 2024
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 functor, examples.
(Slides, pp.6,8-11 and references therein; Pareigis, pp.6-8; Riehl, pp.9-10,13).
Class 3 October 8, 2024
Discussion of homeworks 1-3. Contravariant functor. More examples of functors.
Forgetful functors. Composition of functors is a functor. Recap of
tensor products.
(Slides, pp.12-13 and references therein; Pareigis, pp.6-8; Riehl, pp.13-18).
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
\(S(X)\). Examples of functors \(\mathbf{Group} \to \mathbf{Group}\) sending
a group \(G\) to \(G/Z(G)\), or to \(G/[G,G]\).
There is no functor \(\mathbf{Group} \to \mathbf{Group}\) sendig a group \(G\) to \(Z(G\)). The
category \(\mathbf{Cat}\) of small categories. Isomorphism of categories.
\(\mathbb S_X \simeq \mathbb S_Y\) iff \(|X| = |Y|\).
\(\mathbb C_M \simeq \mathbb C_N\) iff \(M \simeq N\). Isomorphism of objects
within a category. Equivalence of categories (just a definition, motivation and
examples will be considered later).
(Slides, pp.15-16,18,21 and references therein).
Class 5 October 22, 2024
Categories of matrices and finite-dimensional vector spaces are not isomorphic,
but equivalent. Equivalence of categories is an equivalence relation (after
Adamek et al., p.37). 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).
video
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).
video
(mostly disfunctional, with interruptions and without sound)
(referred below as "Slides")
