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Crossed Menagerie

This entry is about the document

• Tim Porter, The Crossed Menagerie: an introduction to crossed gadgetry and cohomology in algebra and topology. (Notes, in part, prepared for the XVI Encuentro Rioplatense de Algebra y Geometría Algebraica, in Buenos Aires, 12-15 December 2006, and for course MATH5312, Spring-Summer term 2007, 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 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.

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

    • Eilenberg-Zilber map, shuffle, products of simplices

    • monoidal Dold-Kan correspondence

• 12 More simplicial constructions!

category: reference