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).
(referred below as "Slides")
