INTRODUCTION TO CATEGORY THEORY
Sunday 16:15-18:00, Room 1.
1. General information
The course is of M.Sc. level. The main aim of the course is to teach the language of category
theory that is widely used in mathematics. If time allows, we will also discuss some
mathematical content of this theory and some applications.
We will discuss the notions of categories and functors important classes of those. We will give
many examples from diﬀerent areas of mathematics.
To understand the main part of material of the course, the only required knowledge is basic
Group theory. However, in order to understand the examples and applications it is
recommended to be familiar with most of the following topics: Linear algebra, Group theory,
Algebraic topology, General topology, Commutative algebra, Diﬀerential Topology,Algebraic
Geometry, Functional analysis.
- Motivation, overview, deﬁnition of category, examples (sets, groups, a group,
abelian groups, modules, posets, a poset, small categories (+2 warnings))
, integers, topology, (pointed) topological spaces, representations, G-sets,
topological vector spaces (Banach, Hilbert, Hilbertian), varieties and manifolds,
metric spaces (isometry/Lipschitz), interesting subcategories of the over categories
(ﬁbration, vector bundle, local systems, (Leray) sheafs), opposite category , initial
and ﬁnal object, commutative diagrams (as a category) over/under cat. (co)limits,
(co)products , ﬁber products, pushout, (co-)equalizer, mono/epi-morphisms.
- Additive categories, examples, (co)ker/(co)image, examples, Abelian categories,
extensions, Grothendieck group.
- Functors, examples (forgetful
functors, duality, (internal) hom, (co)limits (pushforward and pullback between
over categories, Schur functors (tensor/symmetric/exterior powers, determinant)),
pullback of modules and representations), review of diagrams, faithful, full,
essentially surjective, equivalence, examples (NSS, Gelfand Naimark, G-principle
spaces and G, Galois correspondence (for topological spaces), duality of ﬁnite
dimensional spaces, linear spaces and matrices) , criterion for equvivalence.
- Category of functors, Yoneda lemma, (Grothendieck) sheaves, adjoint functor,
examples, review of (co)limits, C-objects in D, examples (group object (top/lie/alg
group), vector bundles, ”application” to commutativity of πn.
- Limits, colimits of categories, localisation of categories, Ore condition, Serre
quotient, localisation, examples.
- Groupoids, examples, classiﬁcation, representations, pull, push, limits, colimits,
- Applications: Van Kampen theorem, Mackey theorem.
- 2-categories, monoidal categories, tannakian categories, ﬁber functor, Deligne
points, monoid and group objects in monoidal categories, cartesian monoidal
categories, cartesian closed categories.
- Exact and continuous functors, relation with adjunction.
projective/injective/compact generators, abelian case.
- Barr-Beck, descent theory.
- Derived functors and categories. Existence and uniqueness criteria.
- Sheaves, sites and topoi.
- Enriched categories, DG categories, Top enriched, model categories, triangulated
- Simplicial sets and objects, Dold-Kan equivalence.
- Nerve, inﬁnity categories.
S. Mac Lane, Categories for the Working Mathematician.
6. E-mail list
To join/un-join the course e-mail list send me an e-mail (from the address you wish to
join/un-join) with subject “join/un-join me to cat-5775”. To send a message to the course
mailing list send me an e-mail with subject “e-mail to cat-5775 – the subject of your
7. How to get credit for the course?
The homework will be 100% of the grade.
8. Lecture notes
To be added