This entry is about the document
This is an ongoing set of notes outlining an approach to nonabelian cohomology, stacks, etc., and Grothendieck’s conjectured extension of ‘Galois-Poincaré theory’. The title refers to the array of strange beasties that occur as generalisations of crossed modules. (The present version is over 921 pages long, and is changing quite often. A fairly stable version (but up-dates are planned shortly!) is available as a download, here.
These notes, or at least the first few chapters of them, acted as course notes for a 4 lecture course in Buenos Aires, and later on for a course on cohomology in Ottawa.
Some idea of the content can be gleaned from the Table of Contents.
(This gives Chapter and section headings. Subsection headings give too long a list to be that useful, so if you want more detail (for the moment) look at the pdf file.)
Introduction
1 Preliminaries
1.1 Groups and Groupoids
1.2 A very brief introduction to cohomology
1.3 Simplicial things in a category
2 Crossed modules - definitions, examples and applications
2.1 Crossed modules
2.2 Group presentations, identities and 2-syzyzgies
2.3 Cohomology, crossed extensions and algebraic 2-types
3 Crossed complexes and (Abelian) Cohomology
3.1 Crossed complexes: the Definition
3.2 Crossed complexes and chain complexes: I
3.5 Simplicial groups and crossed complexes
3.6 Cohomology and crossed extensions
3.7 2-types and cohomology
3.8 Re-examining group cohomology with Abelian coefficients
4 Beyond 2-types
4.1 Crossed squares
4.2 2-crossed modules and related ideas
4.3 Cat$^n$ -groups and crossed $n$-cubes
4.4 Loday’s Theorem and its extensions
4.5 Crossed N-cubes
5 Classifying spaces, and extensions
5.1 Non-Abelian extensions revisited
5.2 Classifying spaces
5.3 Simplicial Automorphisms and Regular Representations
5.4 Simplicial actions and principal fibrations
5.5 $\overline{W}$, $W$, and twisted Cartesian products
5.6 More examples of Simplicial Groups
6 Non-Abelian Cohomology: Torsors, and Bitorsors
6.1 Descent: Bundles, and Covering Spaces
6.2 Descent: simplicial fibre bundles
6.3 Descent: Sheaves
6.4 Descent: Torsors
6.5 Bitorsors
6.6 Relative $M$-torsors
7 Hypercohomology and exact sequences
7.1 Hyper-cohomology
7.2 Mapping cocones and Puppe sequences
7.3 Puppe sequences and classifying spaces
8 Non-Abelian Cohomology: Stacks
8.1 Fibred Categories
8.2 The Grothendieck construction
8.3 Prestacks: sheaves of local morphisms
8.4 From prestacks to stacks
9 Non-Abelian Cohomology: Gerbes
9.1 Gerbes
9.2 Geometric examples of gerbes
9.3 Cocycle description of gerbes
10 Homotopy Coherence and Enriched Categories
10.1 Case study: examples of homotopy coherent diagrams
10.2 Simplicially enriched categories
10.3 Structure
10.4 Nerves and Homotopy Coherent Nerves
10.5 Useful examples
10.6 Two nerves for 2-groups
10.7 Pseudo-functors between 2-groups
11 Other enrichments, other versions of homotopy coherence
11.1 Other enrichments`?
11.2 From simplicially enriched to chain complex enriched
12 More simplicial constructions!