Avraham (Rami) Aizenbud אברהם (רמי) איזנבוד
 Teaching

## INTRODUCTION TO CATEGORY THEORY

Lectures.

Thursday 13:45-15:30, Ziskind 155.

T.A.

Zhuohui Zhang, Sunday 14:15-15:00, Ziskind 155.

Oﬃce hours.

2.  Overview

4.  Topics
5.  Textbook

9.  Homework

### 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.

### 2. Overview

We will discuss the notions of categories and functors important classes of those. We will give many examples from diﬀerent areas of mathematics.

### 3. Prerequisite

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.

### 4. Topics

(1)
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.
(2)
Additive categories, examples, (co)ker/(co)image, examples, Abelian categories, extensions, Grothendieck group.
(3)
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.
(4)
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.
(5)
Limits, colimits of categories, localisation of categories, Ore condition, Serre quotient, localisation, examples.
(6)
Groupoids, examples, classiﬁcation, representations, pull, push, limits, colimits, Groupoid objects.
(7)
Applications: Van Kampen theorem, Mackey theorem.
(8)
2-categories, monoidal categories, tannakian categories, ﬁber functor, Deligne points, monoid and group objects in monoidal categories, cartesian monoidal categories, cartesian closed categories.
(9)
Exact and continuous functors, relation with adjunction. projective/injective/compact generators, abelian case.
(10)
Barr-Beck, descent theory.
(11)
Derived functors and categories. Existence and uniqueness criteria.
(12)
Sheaves, sites and topoi.
(13)
Enriched categories, DG categories, Top enriched, model categories, triangulated categories.
(14)
Simplicial sets and objects, Dold-Kan equivalence.
(15)
Nerve, inﬁnity categories.

### 6. E-mail list

To join/un-join the course e-mail list send Yotam an e-mail (from the address you wish to join/un-join) with subject “join/un-join me to cat-5779”. To send a message to the course mailing list send Yotam an e-mail with subject “e-mail to cat-5779 – the subject of your message”.

### 7. How to get credit for the course?

The homework will be 100% of the grade.