Tim Porter HQFTs meet the Crossed Menagerie

Introduction

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

Remarks by way of explanation

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.

Table of Contents:

1. Crossed modules - definitions, examples and applications

   1. Crossed modules

   2. Group presentations, identities and 2-syzygies

   3. Cohomology, crossed extensions and algebraic 2-types

2. Crossed complexes

   1. Crossed complexes: the Definition

   2. Crossed complexes and chain complexes: I

   3. Associated module sequences

   4. Crossed complexes and chain complexes: II

   5. Simplicial groups and crossed complexes

   6. Cohomology and crossed extensions

   7. 2-types and cohomology

   8. Re-examining group cohomology with Abelian coefficients

3. Syzygies, and higher generation by subgroups

   1. Back to syzygies

   2. A brief sideways glance: simple homotopy and algebraic K-theory

   3. Higher generation by subgroups

   4. Group actions and the nerves

   5. Complexes of groups

   6. Complexes of groups on a scwol

4. Beyond 2-types

   1. $n$-types and decompositions of homotopy types

   2. Crossed squares

   3. 2-crossed modules and related ideas

   4. Cat$^n$-groups and crossed $n$-cubes

   5. Loday’s Theorem and its extensions

   6. Crossed $\mathbb {N}$-cubes

5. Classifying spaces, and extensions

   1. Non-Abelian extensions revisited

   2. Classifying spaces

   3. Simplicial Automorphisms and Regular Representations

   4. Simplicial actions and principal fibrations

   5. $\overline {W}$, $W$ and twisted Cartesian products

   6. More examples of Simplicial Groups

6. Non-Abelian Cohomology: Torsors, and Bitorsors

   1. Descent: Bundles, and Covering Spaces

   2. Descent: simplicial fibre bundles

   3. Descent: Sheaves

   4. Descent: Torsors

   5. Bitorsors

   6. Relative $\mathsf {M}$-torsors

7. Categorifying $G$-torsors and $\mathsf {M}$-torsors

   1. Torsors for a gr-stack

8. Topological (Quantum) Field Theories

   1. What is a topological quantum field theory?

   2. How can one construct TQFTs?

   3. Examples, calculations, etc.

   4. How can one construct TQFTs (continued)?

9. Relative TQFTs: some motivation and some distractions

   1. Beyond TQFTs

   2. Vector bundles and various forms of topological K-theory

   3. Case Study 2: (continued)

10. Homotopy Quantum Field Theories, I

    1. The category of $B$-manifolds and $B$-cobordisms

    2. The definitions of HQFTs

    3. Examples of HQFTs

    4. Change of background

    5. Simplicial approaches to HQFTs.

    6. Formal HQFTs .

11. Algebraic models for 2 dimensional HQFTs

    1. Crossed $\mathsf{C}$-algebras: first steps

    2. From 1+1 HQFTs over $K(G,1)$s to crossed $G$-algebras

    3. Constructions on crossed $G$-algebras

    4. $A$-Frobenius algebras

    5. Crossed $\mathsf{C}$-algebras: the general case

    6. A classification theorem for formal $\mathsf{C}$-HQFTs

    7. Constructions on formal HQFTs and crossed $\mathsf{C}$-algebras

• Index