I’m teaching a course on category theory at U.C. Riverside, and since my website is still suffering from reduced functionality I’ll put the course notes here for now. I taught an introductory course on category theory in 2016, but this one is a bit more advanced.

The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.

• Lecture 1: What is pure mathematics all about? The importance of free structures.

• Lecture 2: The natural numbers as a free structure. Adjoint functors.

• Lecture 3: Adjoint functors in terms of unit and counit.

• Lecture 4: 2-Categories. Adjunctions.

• Lecture 5: 2-Categories and string diagrams. Composing adjunctions.

• Lecture 6: The ‘main spine’ of mathematics. Getting a monad from an adjunction.

• Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

• Lecture 8: The walking monad, the augmented simplex category and the simplex category.

• Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

• Lecture 10: The Dold-Thom theorem: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. The homology of a chain complex.

• Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

• Lecture 8: The walking monad, the

augmented simplex category and the simplex category.

• Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

• Lecture 10: Chain complexes from simplicial abelian groups. The homology of a chain complex.

• Lecture 12: The bar construction: getting a simplicial objects from an adjunction. The bar construction for G-sets, previewed.

• Lecture 13: The adjunction between G-sets and sets.

• Lecture 14: The bar construction for groups.

• Lecture 15: The simplicial set obtained by applying the bar construction to the one-point -set, its geometric realization and the free simplicial abelian group

• Lecture 16: The chain complex coming from the simplicial abelian group its homology, and the definition of group cohomology with coefficients in a -module.

• Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group by an abelian group gives an action of on and a 2-cocycle

• Lecture 18: Classifying abelian extensions of groups. Direct products, semidirect products, central extensions and general abelian extensions. The groups of order 8 as abelian extensions.

• Lecture 19: Group cohomology. The chain complex for the cohomology of with coefficients in , starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of by in terms of

• Lecture 20: Examples of group cohomology: nilpotent groups and the fracture theorem. Higher-dimensional algebra and homotopification: the nerve of a category and the nerve of a topological space. as the nerve of the translation groupoid as the walking space with fundamental group