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Indian Institute of Technology Kanpur

Category Theory

Indian Institute of Technology Kanpur and NPTEL via Swayam

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Overview

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

Syllabus

Week 1:Structure vs. property: monoids, groups, preorders, partial orders; structure preserving maps: homomorphisms, continuous maps; definition and examples of categories; hom-sets and duality; small, locally small, and large categories; set theory vs category theory; Russell’s paradox and the category of all small categories.
Week 2:Functors: covariant and contravariant; skeletons and the axiom of choice; isomorphism; groupoid; monomorphism and epimorphism; full, faithful, essentially surjective functors; natural transformations; equivalence of categories.
Week 3:Building even more categories from the old ones: slice categories and local property, comma categories, functor categories; congruence and quotients; representable functors; Yoneda lemma and Yoneda embedding; separating and detecting families; injective and projective objects; representables are projective.
Week 4:Categorical properties: initial and terminal objects, products, coproducts; diagrams and (co)cones; (co)limits of a given shape; (co)equalizers, regular mono(epi)morphisms; pullbacks and pushouts; direct and inverse limits; constructing all (co)limits from some of them; (co)complete categories; absolute (co)limits; preservation, creation, and reflection of (co)limits.
Week 5:Adjunctions: definition and examples; relation to comma categories; composition of adjoint functors; units, counits, and characterization of adjoint functors; equivalence gives adjunction; right adjoints preserve limits.
Week 6:General adjoint-functor theorem; well-powered categories; special adjoint functor theorem; monads: definition and examples; monads arising from adjunctions.
Week 7:Eilenberg-Moore and Klesili categories; Kleisli isinitial while Eilenberg-Moore is terminal; monadic functors; Beck’s monadicity theorem; Crude monadicity theorem;examples.
Week 8:Introduction to monoidal categories and enriched category theory.
Week 9:Additive and abelian categories
Week 10:Introduction to homological algebra.
Week 11:Introduction to sheaves on topological spaces and locales, and topos theory..
Week 12:Introduction to model categories.

Taught by

Prof. Amit Kuber

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