## March 21, 2008

### 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.)

Posted at March 21, 2008 8:01 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1638

### 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.

Posted by: jim stasheff on March 21, 2008 8:40 PM | Permalink | Reply to this

### Re: Crossed Menagerie

Thanks, Jim.

In the Ottawa MSc lectures, I restricted myself to the group based theory although there were students with a Lie algebra background in the group. My reasoning was that, exactly as you say, the Lie algebra and associative algebra analogues do go through without much pain’ and in fact are sometimes nicer, more useful (e.g., for the L infinity stuff that Urs is interested in), but that if I tried to handle the other areas as well, I risked jumping on my horse and riding off in all directions’, (as one does frequently!) so the students could (and did) investigate those analogies and perhaps that was the best way for them to make the theory there own’ (in modern parlance, to own’ it).

If someone else feels like writing a similar working document (since that is how I see it) for associative algebras, etc., that would be useful.

One question to the blog is how can one do the associative algebra and Lie algebra analogues of the use of classifying spaces. I am sure that the point has been discussed somewhere in the blog and the papers of the contributors to it, but a quick summary could be useful to me.

I will add another comment later. Happy Easter everyone.

Posted by: Tim Porter on March 22, 2008 9:23 AM | Permalink | Reply to this

### Re: Crossed Menagerie

jumping on my horse and riding off in all directions’

that’s known as an explosive departure ;-)

for classifying spaces in hte Lie case at least in characteristic 0 we have the Cartan or Weil models, though not having read your
notes past the intro I’m not sure what you want a classifying space’ to do

and for the assoc alg case, the bar construction

Posted by: jim stasheff on March 22, 2008 1:24 PM | Permalink | Reply to this

### Re: Crossed Menagerie

I’m not sure what you want a classifying space’ to do’

Doh! perhaps to classify something!

Being serious (oh well!) Ronnie and Phil Higgins wrote a paper The classifying space of a crossed complex, Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120,’ and I in my naivity asked What does BC classify?’, thinking of the group case classifying G-bundles.

I think the research is finally getting to give an answer some 18 years later.

Posted by: Tim Porter on March 22, 2008 7:44 PM | Permalink | Reply to this

### Re: Crossed Menagerie

I suppose you know it, but it deserves to be said in this context here again, for those readers who are wondering:

nerves of strict 2-groups $G$ (= crossed modules) classify $G$ 2-bundles. See the work by Baas, Bökstedt and Kro and John Baez and Danny Stevenson on Classifying space of a strict 2-group.

Posted by: Urs Schreiber on March 22, 2008 7:59 PM | Permalink | Reply to this

### Re: Crossed Menagerie

I would add the name of Larry Breen who did something closely related in the Bitorsors paper way back in (?)1992 or Jack Duskin in his paper on Higher Order Descent who looks at Cech cohomology with a 2-group of coefficients.

I, following Dave Yetter also related the classifying space of a finite crossed module to certain TQFTs. (Interpretations of Yetter’s notion of G-coloring : simplicial fibre bundles and non-abelian cohomology, J. Knot Theory and its Ramifications 5 (1996) 687-720)(That was a classic case of research blocked by a referee who said the idea was rubbish and rejected a research grant proposal! The story of my life! Heigh ho.)

Each approach has opened up new uses and insights. The new ones are excellent, but please don’t forget their precursors.

Posted by: Tim Porter on March 22, 2008 9:05 PM | Permalink | Reply to this

### Re: Crossed Menagerie

I would add the name of Larry Breen who did something closely related in the Bitorsors paper way back in (?)1992 or Jack Duskin in his paper on Higher Order Descent who looks at Cech cohomology with a 2-group of coefficients.

My understanding is – but please correct me if I am wrong about this! – that the Baas-Bökstedt-Kro/Baez-Stevenson work is the first to really do the topology.

While it is clear that for $G$ an $n$-group $G$-bundles have cocycles which are given by $n$-functors $g:{Y}^{•}\to BG$ which after hitting them with geometric realization and nerve become maps of topological spaces $\left(\mid {Y}^{•}\mid \to \mid BG\mid \right)\simeq X\to B\mid G\mid$ the hard part is to show the converse, that homotopy classes of maps $X\to B\mid G\mid$ all come from equivalence classes of such $n$-functors. (In fact, Baas-Bökstedt-Kro don’t use transformations of cocycle functors, but “concordances”.)

But John can certainly say more about it, when he finds the time to look at the blog again.

I, following Dave Yetter also related the classifying space of a finite crossed module to certain TQFTs.

I am very fond of that work on the Yetter model you did. I once wrote about it in the entry Dijkgraaf-Witten and its Categorification by Martins and Porter where I enjoyed noticing that the combinatorial weight that enters the “path integral” is precisely the Leinster measure.

Posted by: Urs Schreiber on March 22, 2008 10:45 PM | Permalink | Reply to this

### Re: Crossed Menagerie

Yes, I agree with you, that early work did the simplicial case or with Breen the sheaf theoretic one. Some of the work of the Granada research work had looked at very similar constructions in exact categories, again emphasising the simplicial viewpoint.

The work you mention was the first that took the case of topological 2-groups seriously and that was very important. The problem of handling simplicial topological groups seemed to me to be partially solved by Mostow’s work on smooth spaces in the J. Diff. Geom. that you have recently mentioned, but I never put aside the time to see if my ideas were correct, and seeing the solution from Baas-Bökstedt-Kro/Baez-Stevenson makes me sure that it was only a partial answer that I had glimpsed.

This seems to me, perhaps falsely, to be related to a problem that Turaev and I faced in our Homotopy QFT paper. We used the notion of a formal HQFT, i.e. combinatorially defined with the conditions of an homotopy QFT encoded in that language as well. (That paper is finally to appear. It disappeared into the automatic paper handling machine of a well known journal for a considerable amount of time. It was actually lost, and when they checked, their records did not list it, yet it was on their submissions server. I mention this as a cautionary tale. Alway check on up papers!!!)

I recall you said back in May 2006 (String Coffee Table blog)
“it would be consistent to say that an HQFT for target a K(G,1) is a representation of a category of cobordisms which are equipped with flat G-bundles with connection”,
and that was exactly my intuition when I was working on the paper with Vladimir.

Your further observation was

A 2d formal $C$-HQFT is a representation of the category of 2d-cobordisms which are equipped with (possibly local trivializations of) principal C-2-bundles with 2-connection

and again that was my intuition. The question that was left hanging was whether or not we could remove the word formal’. I felt we could but Vladimir was more cautious and felt my arguments were not watertight.

Has anyone any light they could shed on this?

I was just looking back at my paper on Yetter’s work to reply to another of your posts and it is interesting to note the amount of work I had to do to get the concordance’ or isotopy’ results that showed independence from the triangulation. It is exactly that problem that we could not handle in the HQFT paper, but the topological complications of the more recent case were not there as everything was discrete.

Finally a request for help’: My reason for launching into the Menagerie notes was to try to understand the work of Aldrovandi (2-Gerbes bound by complexes of gr-stacks, and cohomology, arXiv:math/0512453) which seemed to me to be giving a possibly interesting class of background coefficients for the next level of HQFTs. His work was based on an early thesis of Debremaeker using Dedeker’s approach to non-Abelian cohomology.

I am still trying to understand how that stuff interacts with Breen’s approach to 2-gerbes. (I had looked at Aldrovandi’s Dec2005 version, the latest version, still called version 1, is dated Feb 2, 2008, but is not the same as the previous version 1! It looks at Breen’s work but still does not suggest the answer to my question, since that is not thepurpose of the paper.) Has anyone else looked at this?

Posted by: Tim Porter on March 23, 2008 9:38 AM | Permalink | Reply to this

### Re: Crossed Menagerie

Tim quoted:
A 2d formal C-HQFT is a representation of the category of 2d-cobordisms which are equipped with (possibly local trivializations of) principal C-2-bundles with 2-connection

Triggered a possibly irrelevant comment:
The original idea of homology was essentially that of bordism, i.e. thinking of cycles as being submanifolds
but of course that only works up to n = 1 or 2

wonder if the above for 2 fails for 3?

cf also strictification problems in 3

Posted by: jim stasheff on March 23, 2008 4:50 PM | Permalink | Reply to this

### Re: Crossed Menagerie

That suggests looking at the second Turaev paper on HQFTs where he attack dimension 3.(Homotopy field theory in dimension 3 and crossed group-categories,math/0005291). I have not yet really tried to understand it so any helpful comments would be welcome.

Posted by: Tim Porter on March 23, 2008 9:44 PM | Permalink | Reply to this

### Re: Crossed Menagerie

seemed to me to be giving a possibly interesting class of background coefficients for the next level of HQFTs.

[…]

Has anyone else looked at this?

We want in the end QFTs here whose configuration spaces are $\infty$-bundles with connections for a given $\infty$-group. There should be a cocycle on that $\infty$-group which induces a function on this configuration space.

I like to think of this as higher Chern-Simons theory generally, or rather “Dijkgraaf-Witten” theory, if we are in the discrete setup.

In ${L}_{\infty }$-connections we give an algebraic model for higher Chern-Simons bundles obtained from ${L}_{\infty }$-cocycles, essentially noticing that every ${L}_{\infty }$-cocycle $\mu$ on an ${L}_{\infty }$-algebra $g$ which is in transgression with an invariant polynomial $P$ via a transgression element $\mathrm{cs}$ gives rise to a commuting diagram

$\begin{array}{ccc}\mathrm{CE}\left(g\right)& \stackrel{\mu }{←}& \mathrm{CE}\left({b}^{n}u\left(1\right)\right)\\ ↑& & ↑\\ W\left(g\right)& \stackrel{\left(\mathrm{cs},P\right)}{←}& W\left({b}^{n}u\left(1\right)\right)\\ ↑& & ↑\\ \mathrm{inv}\left(g\right)& \stackrel{P}{←}& \mathrm{inv}\left({b}^{n}u\left(1\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$ I am describing in section 5 of Nonabelian differential cohomology # how diagrams like this relate to nonabelian differential cocycles encoding higher bundles with connection. This particular diagram would be the Chern-Simons bundle classified by that cocycle living over $B\mid G\mid$.

General Chern-Simons theory should be a $\Sigma$-model type QFT with

- target space $B\mid G\mid$

- background field that higher Chern-Simons bundle $\mathrm{CS}\to B\mid G\mid$ with connection

- which assigns to a parameter space $\Sigma$ the collection of sections of the result of transgressing $\mathrm{CS}$ to $\mathrm{maps}\left(\Sigma ,B\mid G\mid \right)$.

One needs to find suitable models to realize the structures appearing in this program. I expect that it should be helpful to look directly for the $\infty$-model.

I can’t claim that I am there already, but the notes on Nonabelian differential cohomology building on the ${L}_{\infty }$-connections show how I think the issue can be approached.

That’s maybe not as concrete a reply yet to your “Has anyone else looked at this?”, as you would hope. But I thought I’d mention it.

Posted by: Urs Schreiber on March 24, 2008 12:15 AM | Permalink | Reply to this

### Re: Crossed Menagerie

Tim wrote:

The new ones are excellent, but please don’t forget their precursors.

In some technical sense you can’t forget what you never knew… but your advice is well taken. I really must add a reference to this:

• Brown and Higgins, The classifying space of a crossed complex, Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120.

to my paper with Danny on the classifying space of a topological 2-group. I’m sorry for not doing it earlier, but it’s not too late.

Posted by: John Baez on March 28, 2008 9:10 PM | Permalink | Reply to this

### Re: Crossed Menagerie

how can one do the associative algebra and Lie algebra analogues of the use of classifying spaces

Apart from the discussion of that which we have in ${L}_{\infty }$-connections # I can offer the following integration procedure (as described at the end of Space and Quantity (blog, pdf)):

Let’s say a smooth space is a sheaf on the site whose objects are natural numbers and whose morphisms are smooth maps ${ℝ}^{n}\to {ℝ}^{m}$.

Write $C$ for the category of smooth spaces.

For every smooth space we get a family of $\omega$-groupoids

${\Pi }_{n}:{C}^{\mathrm{op}}\to \omega \mathrm{Cat}$

whose $\left(k-morphisms are thin-homotopy classes of globular $k$-paths and whose $n$-morphisms are full homotopy classes of globular $n$-paths.

Write DGCA for the category of differential ${ℕ}_{+}$-graded commutative algebras. There is a contravariant functor

$S:\mathrm{DGCA}\to C$

(part of an adjunction) which sends each DGCA $A$ to the space

$S\left(A\right):U↦{\mathrm{Hom}}_{\mathrm{DGCA}}\left(A,{\Omega }^{•}\left(U\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

For every finite dimensional ${L}_{\infty }$-algebra on a ${ℕ}_{+}$-graded vector space $g$ we get its Chevalley-Eilenberg DGCA

$\mathrm{CE}\left(g\right):=\left({\wedge }^{•}{g}^{*},{d}_{g}\right)$

with ${d}_{g}$ of degree +1 and $\left({d}_{g}{\right)}^{2}=0$. In fact, this defines the ${L}_{\infty }$-structure on $g$.

Then we can define the strict $n$-group integrating a given ${L}_{\infty }$-algebra $g$ as

$BG:={\Pi }_{n}\left(S\left(\mathrm{CE}\left(g\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

The left hand side means that we have a 1-object $\omega$-groupoid. Indeed, the right hand side always has a single object, as one easily sees (since on the point we only have the 0 differential $\left(p>0\right)$-form).

Let $g$ be an ordinary Lie algebra. Then

$BG:={\Pi }_{1}\left(S\left(\mathrm{CE}\left(g\right)\right)\right)$

is indeed the simply connected Lie group integrating it. The right hand side boils down to nothing but the familiar (in some circles) integration of Lie algebras in terms of classes of Lie algebra valued forms on the interval.

We also have for each $g$ its Weil DGCA $W\left(g\right):=\mathrm{CE}\left(\mathrm{inn}\left(g\right)\right):=\mathrm{CE}\left(\mathrm{Cone}\left(g\stackrel{\mathrm{Id}}{\to }g\right)\right)$ as well as the DGCA of basic forms $\mathrm{inv}\left(g\right):=W\left(g{\right)}_{\mathrm{basic}}$ those invariant under inner derivations along the fiber of the inclusion $g↪\mathrm{inn}\left(g\right)\phantom{\rule{thinmathspace}{0ex}}.$

I think that, for $g$ still an ordinary Lie algebra, the 2-groupoid integrating $\mathrm{inn}\left(g\right)$ is

${\Pi }_{2}\left(S\left(W\left(g\right)\right)\right)\simeq B\left(G\stackrel{\mathrm{Id}}{\to }G\right)$

namely the one-object grouopoid version of the strict 2-group which comes from the crossed module coming from the identity on $G$.

This is what I usually denote

$\left(G\to G\right):=\mathrm{INN}\left(G\right):=EG\phantom{\rule{thinmathspace}{0ex}}.$

So:

- $g$ integrates to $G$

- $\mathrm{inn}\left(g\right)=\mathrm{Cone}\left(g\to g\right)$ integrates to $EG$.

I think this goes through for arbitrary ${L}_{\infty }$-algebras, but then things begin to depend a little more crucially on the model for $\infty$-groupoids which one uses.

For instance, if we stay withing strict $\omega$-groupoids then the general $\omega$-groupoid integrating any ${L}_{\infty }$-algebra would be

$BG={\Pi }_{\omega }\left(S\left(\mathrm{CE}\left(g\right)\right)\right)$

and we’d get the universal $G$-bundle in its $\omega$-groupoid incarnatation as $\begin{array}{cccccc}& BG& \to & BEG& \to & BBG\\ :={\Pi }_{\omega }\circ S\left(& \mathrm{CE}\left(g\right)& ←& W\left(g\right)& ←& \mathrm{inv}\left(g\right)& \right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

But one needs to notice that when doing it this way for $g$ an ordinary Lie algebra, the $\omega$-groupoid $BG$ won’t be equal to a 1-groupoid, but just equivalent. But the benefit is that then $BEG$ exists as an $\omega$-groupoid, too.

I am in the process of preparing notes with more on this.

Posted by: Urs Schreiber on March 22, 2008 3:16 PM | Permalink | Reply to this

### Re: Crossed Menagerie

Thanks for making this available! A vey useful compendium.

So let me see if I can find something about the answer to the question which we were discussing here:

Fix some model of $\infty$-groupoids. Then try to do the following:

for $BG$ a one-object $\infty$-groupoid, form

$BEG:=\mathrm{Cone}\left(BG\stackrel{\mathrm{Id}}{\to }BG\right)$

(where the symbols on the left can be taken to be just notation, but indicate that we have reasons to expect that this is indeed the one-objec $\infty$-groupoid incarnation of the universal $G$-bundle in the same sense that $BG$ is the classifying “space” for some $G$).

Question: what is and how can we make sense of (an approximation to) the one-object $\infty$-groupoid “$BBG$” in general?

You pointed me to the cofiber Puppe sequence, which you mention on p. 190 of your notes, where it is described for spaces.

In that context we should try to form $BG\stackrel{\mathrm{Id}}{\to }BG↪BEG\to BEG/BG$

and try to make sense of the item on the very right.

What I would like to know is:

- how would we define this quotient in our $\infty$-groupoid context?

- what is that quotient in simple cases where the $\infty$-group $G$ we start with is an ordinary group? a strict 2-group? a Gray group?

- how is that quotient related to “$BBG$”?

Actually, I tried to compute this quotient recently not for $\infty$-groups but for ${L}_{\infty }$-algbras. But then I noticed that my technique for computing such quotients of ${L}_{\infty }$-algeba does not apply, because the ${L}_{\infty }$-algebra $g$ is not normal in $\mathrm{Cone}\left(g\to g\right)$ (unless I made a mistake).

After realizing that I concluded to myself that forming this quotient is not the way to go towards forming “$BBG$”. But maybe I am wrong.

As I menioned in the other discussion, the best idea I have, so far, is that “$BBG$” should be the one-object $\infty$-groupoid version of the rational approximation

$\prod _{i}K\left(ℤ,{n}_{i}\right)$

to $B\mid G\mid$, where ${n}_{i}$ is the degree of the $i$th non-trivial rational cohomology group of $B\mid G\mid :=\mid BG\mid$.

Since

$\prod _{i}K\left(ℤ,{n}_{i}\right)=\mid \prod _{i}{B}^{{n}_{i}-1}U\left(1\right)\mid$

I would set

$BBG$$:=B{\prod }_{i}{B}^{{n}_{i}-1}U\left(1\right)$.

As a first consistency check, if $G$ is abelian to start with $G={B}^{k}U\left(1\right)$, then with this definition $BBG$ is ${B}^{k+2}U\left(1\right)$, as it should be.

And indeed, I know that this is a pretty useful approximation, as I described in that other entry:

the sequence

$BG\to BEG\to BBG$

corresponds, if everything is smooth (Lie) to the sequence

$\mathrm{CE}\left(g\right)←W\left(g\right)←\mathrm{inv}\left(g\right)=W\left(g{\right)}_{\mathrm{basic}}$

for the ${L}_{\infty }$-algbra $g$. If we map into these kind of sequences, the term on the left produces a $G$-valued cocycle in nonabelian cohomology, whereas the term on the right produces the corresponding characteristic classes.

So this “rational approximation version of $\mathrm{𝔹𝔹}G$” seems to make good sense.

Does it also follow from a cofiber Puppe sequence construction?

Posted by: Urs Schreiber on March 21, 2008 8:42 PM | Permalink | Reply to this

### Re: Crossed Menagerie

In a later part of the notes, I started trying to unravel what would happen if I started with a non-strict 3-group or rather a sheaf of such. For me that means I have a 3-truncated sheaf of simplicial groups (i.e. the Moore complex is trivial in dimensions 3 or more). That gives a (sheaf of) 2-crossed modules so the interchange law does not hold and is covered by the lifting of the Peiffer commutator of the bottom two terms (as pre-crossed module)

I think that I checked (and also that David Roberts and Urs have done the same sort of calculation), that the next few terms of the cofibre sequence worked well.

The problem from that viewpoint is to construct a truncated simplicial group which when you do the Puppe sequence, the terms you need are somewhere along the way (to the left). Can one do this for $\infty$-groups? I do not know.

With recent comments by Jim, in mind I think that an ${A}_{\infty }$ model of things may be needed. On the other hand a restricted example might be to model things on a $k$-crossed complex. This would correspond in the simplicial case to a simplicial group with the Moore complex and the thin subgroup having trivial intersection in dimensions greater than $k$ (or is it $k+1$, and hence all Whitehead products ending up in those dimensions being trivial. I think this is something like a group objects in $\left(\infty ,k\right)$-categories or some such terminology, but I’m hazy about the current flavour of terminology!

I did this in general i.e. for a morphism of 2-crossed modules, but the details have not been typed up and are an incoherent and unholy mess at the moment so are not to be relied upon.

The task then was to extend Larry Breen’s notion of a torsor over a sheaf of gr-groupoids to this setting, and for the moment I have not attempted this beyond trying to understand Breen’s work on it and to write out detailed proofs of some of the statements that look as if the proof might help my understanding.

Posted by: Tim Porter on March 22, 2008 12:07 PM | Permalink | Reply to this

### Re: Crossed Menagerie

It has just occured to me (so perhaps partially baked not fully formed) that the theory of crossed $n$-cubes and the corresponding $n$-cube complexes may have some useful insights to shed on this problem. There is a filtration of the category of crossed $ℕ$-cubes by reflexive subcategories, and this is reflected (pun intended) back in the category of crossed $n$-complexes. With crossed $ℕ$-cubes, some of the Puppe stuff looks FUN as it is combinatorial and algebraic rather than homotopy theoretic. This may lead nowhere but I have a feeling it may yield other ideas.

(Crossed $n$-cubes are the analogue of cat${}^{n}$-groups i.e. of $n$-fold categories in the category of groups.)

I should also point out that my notes are not full enough when describing the work, in this area, of the Granada research team and Jack Duskin.

Posted by: Tim Porter on March 22, 2008 2:03 PM | Permalink | Reply to this

### Re: Crossed Menagerie

In a later part of the notes, I started trying to unravel what would happen if I started with a non-strict 3-group

I need to read the later parts. I was seeing if you do something like a cofiber Puppe sequence in the context of $n$-groups, not just spaces. When I try to do that, I run into the problem that $X\stackrel{f}{\to }Y\to {C}_{f}$ may not be normal in ${C}_{f}$.

Posted by: Urs Schreiber on March 22, 2008 3:28 PM | Permalink | Reply to this

### Re: Crossed Menagerie

I think my feeling is that you quess the thing you want and build a model of it, then you find a morphism from $Y$ to it with suitable properties, and the test is then does the kernel of the map match with $X$ up to homotopy. I will think about this in your context but don’t guarantee an answer!

Posted by: Tim Porter on March 22, 2008 6:01 PM | Permalink | Reply to this

### Re: Crossed Menagerie

A good example of the lack of info in the rational approximation’ (W Feller remarked that for any number you care about 17 is an approximation). An abelian topological group is a product of K(pi,n)’s but a general infinite loop space need not split at all.
A simple example is given by a 2-stage Postnikov system with k-invariant
K(Z,n) –> K(Z/2,2n) being the cup square.

Posted by: jim stasheff on March 24, 2008 2:52 PM | Permalink | Reply to this

### Re: Crossed Menagerie

So what’s a good approximation to “$\mid BBG\mid$”?

We know that whatever it is it needs to be the home of the characteristic classes. So the rational approximation I had looked allright. But of course it does lack information.

Posted by: Urs Schreiber on March 24, 2008 10:36 PM | Permalink | Reply to this

### Re: Crossed Menagerie

Another thought. If you look at the end of my discussion of the Puppe sequences you will find that for a crossed module $M=\left(C\to P\right)$, you get a nice homotopy exact sequence${\pi }_{1}\left(M\right)\left[1\right]\to M\to {\pi }_{0}\left(M\right).$This goes across without pain to crossed squares, and the advantage there is that you can take the nerve in one direction and get a simplicial group. More or less that says that with a crossed square, the $B$ in the first direction has a straightforward group structure (explicitly given, strict etc.). Thus you can take a second $B$. This gives an instance of a $\mathrm{BBM}$ … and for a crossed $n$-cube that iterates to give the corresponding $\left(n+1\right)$-type is a ${B}^{\left(n\right)}\left(M\right)$. (see Loday’s paper on ${\mathrm{cat}}^{n}$ groups and $\left(n+1\right)$-types).

That short homotopy exact sequence then extends using the $\Omega$ / loops/ fibre Puppe sequence and it may give you what you need. Thus if you can find a big model of the $\mathrm{BBG}$ with the properties you want, and then use the fibre Puppe sequence, it may do the trick.

If an $n$-group is modelled simplicially (possibly by a truncated simplicial group) then the corresponding ${\mathrm{cat}}^{n}$-group gives you an explicit crossed $n$-cube and the above applies. That is quite a round about approach, but it should then be possible to read off a more direct formula for the terms.

The only possible snag is that your theory has been thought out initially without using crossed $n$-cubes so a translation process would seem to be needed if the above sketched’ procedure was to be carried out.

(I’m afraid this is not completely coherent, and I have to catch a ferry at midday so cannot put in the time to double check the idea, but I tried it for crossed squares and the sequence looks very nice. Some of the key constructions are given in my paper :

$n$-Types of Simplicial Groups and Crossed $n$-cubes, Topology, 32, (1993) 5-24.

and are sketched in section 4.4 of the menagerie, but that does not look at the sequence.)

I must rush.

Posted by: Tim Porter on March 25, 2008 8:50 AM | Permalink | Reply to this

### Re: Crossed Menagerie

Another small comment to add to yesterday’s. On the trian across Ireland, I looked at Puppe sequences of simplicial groups, and also some of the green fields passing by. (Then it got dark and I couldn’t see the fields, so had to concentrate on the maths!)

The sequence that I sketch out in the notes is well known, but I remember that the proofs ended up being sketched because they needed quite a bit of Goerss and Jardine. However starting with a fibration of 3-types you get a Puppe fibre sequence (off to the lef) that involves the classifying spaces AND goes on until it gets to ${\Omega }^{2}$ of the codomain.

I thought about calculating that and come up with the double decalage as a way towards it. The neat thing is that would fit very well with the way I think of crossed squares arising from simplicial groups, so I will look (possibly tonight) at that aspect. I will keep you informed of the progress (or lack of it).

Posted by: Tim Porter on March 26, 2008 6:25 PM | Permalink | Reply to this

### Re: Crossed Menagerie

Thanks a lot for the helpful comments! I’ll get back to that in a moment.

Posted by: Urs Schreiber on March 26, 2008 6:46 PM | Permalink | Reply to this

### Re: Crossed Menagerie

I did not get too much time on this since I got back. I did note one thing that may be significant. If you are using ${\pi }_{0}$, then you expect things to be nice in the bottom dimension, but if $G$ is not connected, $\Gamma G$ and $\Omega G$ seem to go a bit astray. They seem the wrong technology.

On the other hand using the decalage, you get the action of ${G}_{0}$ automatically, and a copy of the loops for each connected component and more. You do not loose information until you are ready to do so. You get a fibration gratis and without charge’ but do not need connectedness at all.

I do not know if that helps.

Posted by: Tim Porter on April 1, 2008 5:30 PM | Permalink | Reply to this

### Re: Crossed Menagerie

Tim, thanks!

As you probably have seen, i was a bit distracted by other things. Am hoping to get back to this here tonight.

Actually, I came to think that there must be a way to do this entirely in terms of crossed complexes (so in strict $\infty$-groups), if we allow ourselves to encode a group, say, $G$, not by the obvious crossed complex conctentrated in the lowest degree, but one equivalent to that but possibly with stuff in higher degrees.

The reason is that for the case that our $n$-group is “higher simply connected” this is what I get from integrating Lie $\infty$-algebras:

Every ${L}_{\infty }$-algebra $g$ I can integrate to a crossed complex by forming the $\omega$-groupoid of thin-homotopy classes of globular paths in the classifying space of $g$-valued forms: $BG:={\Pi }_{\omega }\left(S\left(\mathrm{CE}\left(g\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$ This is a one-object $\omega$-groupoid and hence encodes a crossed complex of groups.

The point is that if $g$ is an ordinary Lie algebra, this crossed complex will not be the obvious one $\cdots 1\to 1\to G$ with $G$ the simply connected Lie group integrating $g$ but one equivalent to that (I think). (The standard one would be obtained by using ${\Pi }_{1}$ instead of ${\Pi }_{\omega }$, where we divide out full homotopy at level 1 already.)

The good thing about this is that at the level of ${L}_{\infty }$-algebras I know the sequence that I am looking for, it is:

$\mathrm{CE}\left(g\right)←\mathrm{CE}\left(\mathrm{Cone}\left(g\to g\right)\right)=W\left(g\right)←W\left(g{\right)}_{\mathrm{basic}}$

and the entire sequence integrates to a sequence of $\omega$-groups:

$\begin{array}{ccccc}BG& \to & BEG& \to & BBG\\ {\Pi }_{\omega }\left(S\left(\mathrm{CE}\left(g\right)\right)\right)& \to & {\Pi }_{\omega }\left(S\left(W\left(g\right)\right)\right)& \to & {\Pi }_{\omega }\left(S\left(\mathrm{inv}\left(g\right)\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

This suggests that given any crossed complex/strict one-object $\infty$-groupoid $BG$, there one equivalent to it such that $BEG$ and “$BBG$” exist as crossed complexes.

Posted by: Urs Schreiber on April 2, 2008 6:49 AM | Permalink | Reply to this

Post a New Comment