This entry is about the document
University of Ottawa) (pdf)
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 1004 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 an indication of the Chapter and section headings, although these have changed in detail for the latest version available. 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 Syzygies, and higher generation by subgroups
5 Beyond 2-types
5.1 Crossed squares
5.2 2-crossed modules and related ideas
5.3 Cat$^n$ -groups and crossed $n$-cubes
5.4 Loday’s Theorem and its extensions
5.5 Crossed N-cubes
6 Classifying spaces, and extensions
6.1 Non-Abelian extensions revisited
6.2 Classifying spaces
6.3 Simplicial Automorphisms and Regular Representations
6.4 Simplicial actions and principal fibrations
6.5 $\overline{W}$, $W$, and twisted Cartesian products
6.6 More examples of Simplicial Groups
7 Non-Abelian Cohomology: Torsors, and Bitorsors
7.1 Descent: Bundles, and Covering Spaces
7.2 Descent: simplicial fibre bundles
7.3 Descent: Sheaves
7.4 Descent: Torsors
7.5 Bitorsors
7.6 Relative $M$-torsors
8 Hypercohomology and exact sequences
8.1 Hyper-cohomology
8.2 Mapping cocones and Puppe sequences
8.3 Puppe sequences and classifying spaces
9 Non-Abelian Cohomology: Stacks
9.1 Fibred Categories
9.2 The Grothendieck construction
9.3 Prestacks: sheaves of local morphisms
9.4 From prestacks to stacks
10 Non-Abelian Cohomology: Gerbes
10.1 Gerbes
10.2 Geometric examples of gerbes
10.3 Cocycle description of gerbes
11 Homotopy Coherence and Enriched Categories
11.1 Case study: examples of homotopy coherent diagrams
11.2 Simplicially enriched categories
11.3 Structure
11.4 Nerves and Homotopy Coherent Nerves
11.5 Useful examples
11.6 Two nerves for 2-groups
11.7 Pseudo-functors between 2-groups
12 Other enrichments, other versions of homotopy coherence
12.1 Other enrichments`?
12.2 From simplicially enriched to chain complex enriched
13 More simplicial constructions!
14 Indexed / weighted limits and colimits
(This gives the list for the first few chapters. There are more, but they are in a more fluid state. Anyone who would like a `fluid copy' is welcome to contact the author.)
Last revised on September 24, 2019 at 13:00:55. See the history of this page for a list of all contributions to it.