# Category Theory

- Institution: IIT Kanpur
- Department: Mathematics
- Teacher: Prof. Amit Kuber
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111104524)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111104524)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

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

1. [Lecture 01: Introduction to category theory](https://invisiblenetwork.ai/learn/nptel/111104524/d49a3502cca0) (YouTube)
2. [Lecture 02: Examples of categories](https://invisiblenetwork.ai/learn/nptel/111104524/c0e22261f686) (YouTube)
3. [Lecture 03: Functors](https://invisiblenetwork.ai/learn/nptel/111104524/30c624c7bafd) (YouTube)
4. [Lecture 04: Natural transformations and equivalences of categories](https://invisiblenetwork.ai/learn/nptel/111104524/416149613573) (YouTube)
5. [Lecture 05: Equivalence of categories and some properties of morphisms](https://invisiblenetwork.ai/learn/nptel/111104524/db1c72b75af8) (YouTube)
6. [Lecture 06: The Yoneda lemma and Representable functors](https://invisiblenetwork.ai/learn/nptel/111104524/d0cc7550d5b5) (YouTube)
7. [Lecture 7: Limits and colimits-I: Definitions and examples](https://invisiblenetwork.ai/learn/nptel/111104524/df0565a9e9d5) (YouTube)
8. [Lecture 8: Limits and colimits-II: Constructing new (co)limits from existing (co)limits](https://invisiblenetwork.ai/learn/nptel/111104524/a48007b86dc7) (YouTube)
9. [Lecture 9: Limits and colimits-III: Interaction with functors, Adjunctions-I: Definition](https://invisiblenetwork.ai/learn/nptel/111104524/bf3a08b1eb34) (YouTube)
10. [Lecture 10: Adjunctions-II: Examples](https://invisiblenetwork.ai/learn/nptel/111104524/a6cba2ece896) (YouTube)
11. [Lecture 11: Adjunctions-III: Triangular Identities](https://invisiblenetwork.ai/learn/nptel/111104524/aedeb2e94b95) (YouTube)
12. [Lecture 12: Adjunctions-IV: General adjoint functor theorem](https://invisiblenetwork.ai/learn/nptel/111104524/b755c4b6af21) (YouTube)
13. [Lecture 13: Adjunctions-V: Special adjoint functor theorem, Filtered Colimits-I: Basics](https://invisiblenetwork.ai/learn/nptel/111104524/9de2d8a515f4) (YouTube)
14. [Lecture 14: Filtered colimits-II: Locally finitely presentable (LFP) categories](https://invisiblenetwork.ai/learn/nptel/111104524/6a39d76d6785) (YouTube)
15. [Lecture 15: Monads-I: Eilenberg-Moore and Kleisli categories](https://invisiblenetwork.ai/learn/nptel/111104524/e21dbd1722fe) (YouTube)
16. [Lecture 16: Monads-II: Monadicity theorem](https://invisiblenetwork.ai/learn/nptel/111104524/d60ce244a309) (YouTube)
17. [Lecture 17: Monoidal categories and enriched categories](https://invisiblenetwork.ai/learn/nptel/111104524/b546d471d2f9) (YouTube)
18. [Lecture 18: Abelian categories](https://invisiblenetwork.ai/learn/nptel/111104524/e9f324c10387) (YouTube)
19. [Lecture 19: Grothendieck categories and localization](https://invisiblenetwork.ai/learn/nptel/111104524/228ff255a3f3) (YouTube)
20. [Lecture 20: Freyd-Mitchell embedding theorem, Homological algebra I: Diagram chasing](https://invisiblenetwork.ai/learn/nptel/111104524/c43f87972cc3) (YouTube)
21. [Lecture 21: Homological algebra II: Chain homotopy, projective resolutions and the derived category](https://invisiblenetwork.ai/learn/nptel/111104524/33f2514bba55) (YouTube)
22. [Lecture 22: Model categories](https://invisiblenetwork.ai/learn/nptel/111104524/14765c22245d) (YouTube)
23. [Lecture 23: Topos theory I: Elementary toposes](https://invisiblenetwork.ai/learn/nptel/111104524/07ad9f08a029) (YouTube)
24. [Lecture 24: Topos theory II: Grothendieck toposes](https://invisiblenetwork.ai/learn/nptel/111104524/05632f009e46) (YouTube)