The aim of these notes is to provide some background material for discussions of Homotopy Quantum Field Theories, crossed Frobenius Algebras, simplicial group methods, algebraic models for n-types, crossed modules, etc
These are notes prepared for a minicourse at the Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity, Lisbon, February, 2011. They were constructed from the main body of the much larger Menagerie notes. The method used to prepare them has been to delete sections that were not more or less necessary for this course and then to add in new material. (There will be some loose ends therefore, and missing links. These are given with a ? as the Latex referred to the original cross reference.)
There are several points to make. As in the full Menagerie notes, there are no exercises as such, but at various points if a proof could be expanded, or is left to the reader, then, yes, bold face will be used to suggest that that is a useful place for more input from the reader. In lots of places, reading the details is not that efficient a way of getting to grips with the calculations and ideas, and there is no substitute for doing it yourself. That being said guidance as to how to approach the subject will often be given.
Crossed modules - definitions, examples and applications
Crossed modules
Group presentations, identities and 2-syzygies
Cohomology, crossed extensions and algebraic 2-types
Crossed complexes
Crossed complexes: the Definition
Crossed complexes and chain complexes: I
Associated module sequences
Crossed complexes and chain complexes: II
Simplicial groups and crossed complexes
Cohomology and crossed extensions
2-types and cohomology
Re-examining group cohomology with Abelian coefficients
Syzygies, and higher generation by subgroups
Back to syzygies
A brief sideways glance: simple homotopy and algebraic K-theory
Higher generation by subgroups
Group actions and the nerves
Complexes of groups
Complexes of groups on a scwol
Beyond 2-types
$n$-types and decompositions of homotopy types
Crossed squares
2-crossed modules and related ideas
Cat$^n$-groups and crossed $n$-cubes}{183}{section.4.4}
Loday’s Theorem and its extensions}{189}{section.4.5}
Crossed $\mathbb {N}$-cubes}{196}{section.4.6}
Classifying spaces, and extensions
Non-Abelian extensions revisited
Classifying spaces
Simplicial Automorphisms and Regular Representations
Simplicial actions and principal fibrations
$\overline {W}$, $W$ and twisted Cartesian products
More examples of Simplicial Groups
Non-Abelian Cohomology: Torsors, and Bitorsors
Descent: Bundles, and Covering Spaces
Descent: simplicial fibre bundles
Descent: Sheaves
Descent: Torsors
Bitorsors
Relative $\mathsf {M}$-torsors
Categorifying $G$-torsors and $\mathsf {M}$-torsors
Topological (Quantum) Field Theories
What is a topological quantum field theory?
How can one construct TQFTs?
Examples, calculations, etc.
How can one construct TQFTs (continued)?
Relative TQFTs: some motivation and some distractions
Beyond TQFTs
Vector bundles and various forms of topological K-theory
Case Study 2: (continued)
Homotopy Quantum Field Theories, I
The category of $B$-manifolds and $B$-cobordisms
The definitions of HQFTs
Examples of HQFTs
Change of background
Simplicial approaches to HQFTs.
Formal HQFTs .
Algebraic models for 2 dimensional HQFTs
Crossed $\mathsf{C}$-algebras: first steps
From 1+1 HQFTs over $K(G,1)$s to crossed $G$-algebras
Constructions on crossed $G$-algebras
$A$-Frobenius algebras
Crossed $\mathsf{C}$-algebras: the general case
A classification theorem for formal $\mathsf{C}$-HQFTs
Constructions on formal HQFTs and crossed $\mathsf{C}$-algebras