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).
(referred below as "Slides")
