### Crossed Menagerie

#### Posted by Urs Schreiber

Tim Porter kindly made the following notes available online:

Tim Porter

Crossed Menagerie:an introduction to crossed gadgetry and cohomology in algebra and topology

(pdf with the first 7 chapters (237 pages))

**Intoduction**

These notes were originally intended to supplement lectures given at the Buenos Aires meeting in December 2006, and have been extended to give a lot more background for a course in cohomology at Ottawa (Summer term 2007). They introduce some of the family of crossed algebraic gadgetry that have their origins in combinatorial group theory in the 1930s and ’40s, then were pushed much further by Henry Whitehead in the papers on Combinatorial Homotopy, in particular, [113].

Since about 1970, more information and more examples have come to light, initially in the work of Ronnie Brown and Phil Higgins, (for which a useful central reference will be the forthcoming, [29]), in which crossed complexes were studied in depth. Explorations of crossed squares by Loday and Guin-Valery, [64, 81] and from about 1980 onwards indicated their relevance to many problems in algebra and algebraic geometry, as well as to algebraic topology have become clear. More recently in the guise of 2-groups, they have been appearing in parts of differential geometry, [21, 10] and have, via work of Breen and others, [17, 18, 19, 20], been of central importance for non-Abelian cohomology. This connection between the crossed menagerie and non-Abelian cohomology is almost as old as the crossed gadgetry itself, dating back to Dedecker’s work in the 1960s, [48]. Yet the basic message of what they are, why they work, how they relate to other structures, and how the crossed menagerie works, still need repeating, especially in that setting of non-Abelian cohomology in all its bewildering beauty.

The original notes have been augmented by additional material, since the link with non-Abelian cohomology was worth pursuing in much more detail. These notes thus contain an introduction to the way ‘crossed gadgetry’ interacts with non-Abelian cohomology and areas such as topological and homotopical quantum field theory. This entails the inclusion of a fairly detailed introduction to torsors, gerbes etc. This is based in part on Larry Breen’s beautiful Minneapolis notes, [20].

If this is the first time you have met this sort of material, then some words of warning and welcome are in order.

There is much too much in these notes to digest in one go!

There is probably a lot more than you will need in your continuing research. For instance, the material on torsors, etc., is probably best taken at a later sitting and the chapter ‘Beyond 2-types’ is not directly used until a lot later, so can be glanced at.

I have concentrated on the group theoretic and geometric aspects of cohomology, since the non-Abelian theory is better developed there, but it is easy to attack other topics such as Lie algebra cohomology, once the basic ideas of the group case have been mastered and applications in differential geometry do need the torsors, etc. I have emphasised approaches using crossed modules (of groups). Analogues of these gadgets do exist in the other settings (Lie algebras, etc.), and most of the ideas go across without too much pain. If handling a non-group based problem (e.g. with monoids or categories), then the internal categorical aspect - crossed module as internal category in groups - would replace the direct method used here. Moreover the group based theory has the advantage of being central to both algebraic and geometric applications.

The aim of the notes is not to give an exhaustive treatment of cohomology. That would be impossible. If at the end of reading the relevant sections the reader feels that they have some intuition on the meaning and interpretation of cohomology classes in their own area, and that they can more easily attack other aspects of cohomological and homotopical algebra by themselves, then the notes will have succeeded for them.

Although not ‘self contained’, I have tried to introduce topics such as sheaf theory as and when necessary, so as to give a natural development of the ideas. Some readers will already have been introduced to these ideas and they need not read those sections in detail. Such sections are, I think, clearly indicated. They do not give all the details of those areas, of course. For a start, those details are not needed for the purposes of the notes, but the summaries do try to sketch in enough ‘intuition’ to make it reasonable clear, I hope, what the notes are talking about! (This version is a shortened version of the notes. It does not contain the material on gerbes. It is still being revised. The full version will be made available later.)

## Re: Crossed Menagerie

Just one slight gloss on these marvelous notes

For some of us, interpreting things in terms of Lie algebras (infinitesimal crossed modules) or associative algebras works just as well.