Course pages 2018–19

# Category Theory

## Lecture slides

**Lecture 1**(9 Oct) Introduction; some history; content of this course. Definition of category. The category of sets and functions. Commutative diagrams. Alternative definitions of category.**Lecture 2**(11 Oct) Examples of categories: pre-ordered sets and monotone functions; monoids and monoid homomorphisms; a pre-ordered set as a category; a monoid as a category. Definition of isomorphism.**Lecture 3**(16 Oct) Terminal objects. The opposite of a category and the duality principle. Initial objects. Free monoids as initial objects.**Lecture 4**(18 Oct) Binary products and coproducts.**Lecture 5**(23 Oct) Exponential objects: in the category of sets and in general. Cartesian closed categories: definition and examples.**Lecture 6**(25 Oct) Intuitionistic Propositional Logic (IPL) in Natural Deduction style. Semantics of IPL in a cartesian closed pre-ordered set.**Lecture 7**(30 Oct) Simply Typed Lambda Calculus (STLC). Semantics of STLC types in a cartesian closed category (ccc).**Lecture 8**(1 Nov) Semantics of STLC terms as morphisms in a ccc. Capture-avoiding substitution and its semantics in a ccc.**Lecture 9**(6 Nov) Soundness of the ccc semantics for beta-eta equality of lambda terms. The internal language of a ccc. Free cccs. The Curry-Howard-Lawvere/Lambek correspondence.**Lecture 10**(8 Nov) Functors. Contravariance. Identity and composition for functors. Size: small categories and locally small categories. The category of small categories.**Lecture 11**(13 Nov) Natural transformations. Functor categories. The category of small categories is cartesian closed.**Lecture 12**(15 Nov) Adjunctions. Examples of adjoint functors. Hom functors. Natural isomorphisms.**Lecture 13**(20 Nov) Theorem characterizing the existence of right (respectively left) adjoints in terms of a universal property.**Lecture 14**(22 Nov) Dependent types. Dependent product sets and dependent function sets as adjoint functors.**Lecture 15**(27 Nov) Presheaf categories. The Yoneda lemma.**Lecture 16**(29 Nov) The Yoneda functor is full and faithful. Presheaf categories are cartesian closed.

## Exercise sheets

**Exercise Sheet 1****Exercise Sheet 2****Exercise Sheet 3****Exercise Sheet 4****(GRADED; worth 25% of final mark; hand solutions in to Graduate Education Office by 16:00 on Thursday 15 November.)****Exercise Sheet 5****Exercise Sheet 6**

## Office hours

- The module lecturer will be available to answer questions about the
course material and exercises
**on Fridays between 9:00 and 10:00 in FC08 during Full Term**.

## Additional material

- The following is the classic text on category theory and is
definitely worth looking at if you are feeling mathematically
mature:
Mac Lane, Saunders.

*Categories for the Working Mathematician*. Graduate Texts in Mathematics 5, second ed. (Springer, 1988), ISBN 0-387-98403-8. - A student-oriented guide to on-line material on Category Theory is available at http://www.logicmatters.net/categories/.
- The Category Theory Seminar is held in Centre for Mathematical Sciences.
- Julia Goedecke's Category Theory course for the 2013/4 Mathematical Tripos Part III / Masters of Mathematics.
- Category Theory in the nLab.
- The Catsters' Category Theory Videos.
- Eugenia Cheng's book on
*Cakes, Custard and Category Theory*(published in the US under the title "How to Bake Pi"). - David Spivak's book on
*Category Theory for the Sciences*.