I taught an introductory course on category theory Winter 2016 but this one is a bit more advanced, and less well organized. It focuses on the bar construction for monads, group cohomology, and their connection to categorification and 'homotopification'.

Here are hand-written notes for each class taken by Christian Williams. If you discover any errors in the notes please email me, and I'll add them to the list of errors.

- 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: Chain complexes from simplicial abelian groups. The homology of a chain complex.
- Lecture 11: Comonads from adjunctions. The walking comonad. The bar construction.
- 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 \( \mathbb{E}G \) obtained by applying the bar construction to the one-point \(G\)-set, its geometric realization \( EG = |\mathbb{E}G| \), and the free simplicial abelian group \( \mathbb{Z}[\mathbb{E}G] \).
- Lecture 16: The chain complex \( C(G) \) coming from the simplicial abelian group \( \mathbb{Z}[\mathbb{E}G] \), its homology, and the definition of group cohomology \(H^n(G,A)\) with coefficients in a \(G\)-module.
- Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group \(G\) by an abelian group \(A\) gives an action of \(G\) on \(A\) and a 2-cocycle \(c \colon G^2 \to A\).
- 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: A 2-cocycle in everyday life: the carry digit. The chain complex for the cohomology of \(G\) with coefficients in \(A\), starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of \(G\) by \(A\) in terms of \(H^2(G,A)\).
- 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. \(\mathbb{E}G\) as the nerve of the translation
groupoid \(G/\!/G\). \(BG = EG/G\) as the walking space with fundamental
group \(G\).
- Lecture 21: Homotopification
and higher algebra. Internalizing concepts in categories with finite
products. Pushing forward internalized structures using functors
that preserve finite products. Why the 'discrete category on a set' functor
\(\mathrm{Disc} \colon \mathrm{Set} \to \mathrm{Cat}\),
the 'nerve of a category' functor
\(\mathrm{N} \colon \mathrm{Cat} \to \mathrm{Set}^{\Delta^{\mathrm{op}}}\),
and the 'geometric realization of a simplicial set' functor
\(|\cdot| \colon \mathrm{Set}^{\Delta^{\mathrm{op}}} \to \mathrm{Top}\)
preserve products.
- Lecture 22: Monoidal
categories. Strict monoidal categories as monoids in \(\mathrm{Cat}\)
or one-object 2-categories. The periodic table of strict
\(n\)-categories. General 'weak' monoidal categories.
- Lecture 23: 2-Groups. The periodic table of weak \(n\)-categories. The stabilization hypothesis. The homotopy hypothesis. Classifying 2-groups with \(G\) as the group of objects and \(A\) as the abelian group of automorphisms of the unit object in terms of \(H^3(G,A)\). The Eckmann–Hilton argument.

You can also get hand-written notes by Kenny Courser for the lectures starting with Lecture 8 as a single file.

baez@math.removethis.ucr.andthis.edu

© 2018 John Baez