## 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.

baez@math.removethis.ucr.andthis.edu
© 2018 John Baez