Seminar - Fall 2018

Category Theory

John Baez

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.