Category Theory

By Prof. Amit Kuber | IIT Kanpur

Learners enrolled: 389 | Exam registration: 15

ABOUT THE COURSE:
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.

Summary

Course Status :	Ongoing
Course Type :	Elective
Duration :	12 weeks
Category :	Mathematics
Credit Points :	3
Level :	Undergraduate/Postgraduate
Start Date :	22 Jul 2024
End Date :	11 Oct 2024
Enrollment Ends :	05 Aug 2024
Exam Registration Ends :	16 Aug 2024
Exam Date :	27 Oct 2024 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1: Motivation: sets vs categories, internal vs external, category theory as a unifying language, universal properties; structure vs. properties, structure preserving maps; Categories: definition and examples; hom-sets, locally small and small categories, Russell's paradox; dual and duality principle; initial and terminal objects: definition and examples; isomorphisms and groupoids.

Week 2: Functors: definition and examples, covariant and contravariant, forgetful and free; congruence and quotient categories; natural transformation: definition and examples, natural isomorphism; equivalence of categories; full, faithful and essentially surjective functors.

Week 3: Characterisation of equivalence of categories, skeletal categories; (split) monomorphism and epimorphisms, dimorphism’s and balanced categories; representable functors: definition and examples; Yoneda lemma; universal elements of representable functors; Yoneda embedding.

Week 4: Equalisers and regular mono/epi; building more categories from the old ones: (co)slice categories, arrow categories; (co)limits: shapes and diagrams, definition of (co)limits, uniqueness of (co)limits; types of limits: terminal, pullbacks and products, inverse limits; types of colimits: initial, coequalizer and pushouts, direct limits; constructing all limits from products and equalisers, all finite limits from pullbacks and terminal.

Week 5: (co)complete categories; absolute (co)limits; preservation, creation, and reflection of (co)limits; (co)limits in functor categories; adjoint pairs of functors: definitions and examples including Hom-tensor, free-forgetful, Galois connections, (co)reflections, self-adjoint functors; characterisation of right adjoints in terms of initial objects in certain arrow categories.

Week 6: Uniqueness of left/right adjoints; composition of adjunctions, right adjoints preserve limits; horizontal and vertical compositions of natural transformations and functors; units and counits of adjunctions; characterisation of adjunctions using triangular identities; counit of reflection; equivalence gives adjoint equivalence; adjoint functor theorem: primeval version, solution set condition, general version: (non) examples.

Week 7: Special adjoint functor theorem: coseparators, subobjects and well-powered categories; filtered categories and filtered colimits, commutativity of certain limits and colimits, finitely presentable objects, locally finitely presentable category, Gabriel-Ulmer duality.

Week 8: Monads: definition and examples, monads arising from adjunctions, Eilenberg-Moore and Kleisli categories; Kleisli is initial while Eilenberg-Moore is terminal in the category of adjunctions over a monad, monadic functors, Beck’s monadicity theorem: precise and crude versions, examples.

Week 9: Monoidal categories: definition and examples, strict monoidal functors, types of monoidal categories: symmetric, closed and cartesian closed categories, enriched categories: definition and examples; additive categories: zero objects, biproducts, (co)kernels, abelian categories, image factorisation, properties of abelian categories

Week 10: Exact sequences and (left/right) exact functors; Grothendieck categories and their properties; localisation of categories, Gabriel-Zisman multiplicative systems, Serre subcategories, Gabriel-Popescu theorem, Freyd-Mitchell embedding theorem; Homological algebra: chain complexes, chain maps, five lemma, snake lemma, homology.

Week 11: Long exact sequence of homology, chain homotopy, homotopy category and derived category, homology is homotopy-invariant; projective resolutions, horseshoe lemma, derived functors; Model categories: motivation, weak factorisation systems, functorial factorisation, model structures: definition and examples: chain complexes, Quillen model structure on topological spaces, Quillen adjunction and homotopy categories, introduction to small object argument.

Week 12: Topos theory: (pre)sheaves over a topological space; elementary toposes: subobject classifier, exponentials, image factorisation, internal and external Heyting algebras, examples of elementary toposes; sieves, Grothendieck topology: definition and examples, frames and locales, Grothendieck toposes as sheaves on a site, Giraud’s theorem.

Books and references

1. S. MacLane, Categories for the working mathematician, Vol. 5, Springer Science & Business Media, 2013.
2. S. Awodey, Category theory, Oxford University Press, 2010.
3. M. Kashiwara and P. Schapira, Categories and sheaves, Vol. 332, Springer Science & Business Media, 2005.
4. J. Ada’mek, H. Herrlich and G. E. Strecker, Abstract and concrete categories–The joy of cats, 2004.
5. F. Borceaux, Handbook of categorical algebra 1: Basic category theory, Vol. 50, Encyclopedia of Mathematics and its Applications, 1994.

Instructor bio

Prof. Amit Kuber

IIT Kanpur

Prof. Amit Kuber is a faculty member at the Department of Mathematics and Statistics, IIT Kanpur. He obtained his Ph.D. in Mathematical Logic in 2014 from University of Manchester. His current research interests are representation theory of associative algebras, model theory and category theory.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 27 October 2024 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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