Avraham (Rami) Aizenbud

אברהם (רמי) איזנבוד

Lectures.

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

T.A.

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

Office hours.

By appointment.

Contents

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 π_n.

(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.

5. Textbook

S. Mac Lane, Categories for the Working Mathematician.

6. E-mail list

To join/un-join the course e-mail list send Zhuohui Zhang 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 Zhuohui 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.

8. Lecture notes

To be added

9. Homework