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Category Theory

Prof. Amit Kuber · IIT Kanpur

Start with lecture 1 At NPTEL

About

Category theory is a foundation of mathematics that tries to study structures from an extrinsic viewpoint, i.e., in relation with other structures of similar kinds, just like a society! The internal structure of objects in a category (e.g., elements in a set) is irrelevant if it is not visible through morphisms. Most of the course will cover all standard concepts like (co)limits, adjunction, monads, and the Yoneda lemma, and we will look at introductions to different branches of category theory at the end. The course has been designed to emphasize on similarities and differences between different areas of mathematics and is also useful for people interested in type theory, functional programming (Computer Science) and quantum field theories (Theoretical Physics). I can ensure that this course will change your language and view towards mathematics!

Intended audience: Advanced undergraduates or postgraduates from Mathematics/Computer Science or Theoretical Physics

Prerequisites: None but mathematical maturity is necessary. For understanding and appreciating examples, domain-specific knowledge would be required.

Lectures

Week 01

  1. Lecture 01: Introduction to category theory YouTube
  2. Lecture 02: Examples of categories YouTube

Week 02

  1. Lecture 03: Functors YouTube
  2. Lecture 04: Natural transformations and equivalences of categories YouTube

Week 03

  1. Lecture 05: Equivalence of categories and some properties of morphisms YouTube
  2. Lecture 06: The Yoneda lemma and Representable functors YouTube

Week 04

  1. Lecture 7: Limits and colimits-I: Definitions and examples YouTube
  2. Lecture 8: Limits and colimits-II: Constructing new (co)limits from existing (co)limits YouTube

Week 05

  1. Lecture 9: Limits and colimits-III: Interaction with functors, Adjunctions-I: Definition YouTube
  2. Lecture 10: Adjunctions-II: Examples YouTube

Week 06

  1. Lecture 11: Adjunctions-III: Triangular Identities YouTube
  2. Lecture 12: Adjunctions-IV: General adjoint functor theorem YouTube

Week 07

  1. Lecture 13: Adjunctions-V: Special adjoint functor theorem, Filtered Colimits-I: Basics YouTube
  2. Lecture 14: Filtered colimits-II: Locally finitely presentable (LFP) categories YouTube

Week 08

  1. Lecture 15: Monads-I: Eilenberg-Moore and Kleisli categories YouTube
  2. Lecture 16: Monads-II: Monadicity theorem YouTube

Week 09

  1. Lecture 17: Monoidal categories and enriched categories YouTube
  2. Lecture 18: Abelian categories YouTube

Week 10

  1. Lecture 19: Grothendieck categories and localization YouTube
  2. Lecture 20: Freyd-Mitchell embedding theorem, Homological algebra I: Diagram chasing YouTube

Week 11

  1. Lecture 21: Homological algebra II: Chain homotopy, projective resolutions and the derived category YouTube
  2. Lecture 22: Model categories YouTube

Week 12

  1. Lecture 23: Topos theory I: Elementary toposes YouTube
  2. Lecture 24: Topos theory II: Grothendieck toposes YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Kanpur
Department
Mathematics
Teacher
Prof. Amit Kuber
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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