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 808 pages long, and is changing regularly. 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!