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March 13, 2008

Slides: On Nonabelian Differential Cohomology

Posted by Urs Schreiber

On nonabelian differential cohomology
(52 pdf slides)

This is supposed to be one way to motivate the definition of L L_\infty-connections (pdf, blog, arXiv) along the lines discussed in (Generalized) Differential Cohomology and Lie Infinity-Connections.


Some topics are only mentioned rather briefly in these slides:

For more on the smooth spaces and smooth classifying spaces for L L_\infty-valued forms, see Space and Quantity.

For more on the nature of EG:=INN 0(G)\mathbf{E}G := \mathrm{INN}_0(G) see The inner automorphism 3-group of a strict 2-group.

For more on the functorial description of connections used here, see The first edge of the cube.

For more on the relation between smooth 2-functors and Lie 2-algebra valued forms, see Smooth 2-functors and differential forms.


Question on simplicial nn-categories.

In view of the discussion around slide 32, here is something I should try to better understand:

(here is pdf with more details on this question).

given a simplicial category, what’s the canonical procedure which produces from it a plain category “encoding the same information”.

More concretely:

given a space XX and a regular epimorphism π:YX\pi : Y \to X, we get the simlicial space

Y [3]Y [2]Y. \cdots Y^{[3]} \stackrel{\stackrel{\to}{\to}}{\to} Y^{[2]} \stackrel{\to}{\to} Y \,.

Thinking of YY as a discrete category, this is a simplicial category. But the category I would be after in this simple case is just the Čech groupoid, which is the pair groupoid (codiscrete groupoid) over YY.

Here its clear what’s going on: the original simplicial set is just the nerve of the Čech groupoid.

But now pick some notion P(X)P(X) of groupoid of paths in a space XX. Then we get a genuine simplicial category

P(Y [3])P(Y [2])P(Y). \cdots P(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} P(Y^{[2]}) \stackrel{\to}{\to} P(Y) \,.

I want to form something like the “weak coequalizer”

P 1 π(X) P_1^\pi(X)

of this. The concrete description of what I mean by that is in definition 2.11 here. In appendix A.1 of that we describe the universal property of this construction.

This is the groupoid which is generated from paths in YY and the “jumps between fibers” known from the Čech groupoid, modulo some essentially obvious relations.

Here my question is: what is it we are really doing there? I am thinking that I missed some general nonsense which should, when identified, make all this come out more automatically.

That P 1 π(X)P_1^\pi(X) is essentially the weak coequalizer of

P 1(Y [2])P 1(Y) P_1(Y^{[2]}) \stackrel{\to}{\to} P_1(Y)

but subject to the constraint that the 2-morphism appearing (due to it being a weak coequalizer) in a sense coequalizes

P 1(Y [3])P 1(Y [2]). P_1(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} P_1(Y^{[2]}) \,.

That last part of the sentence is at best vague. And that’s the reason for my question: what general abstract construction is lurking here in the background?

I had addressed that issue quite a while back originally in an entry called Universal Transition, but back then I didn’t mention the term simplicial category, which is probably necessary to ring a bell here with anyone.

And of course then the next step is to do the same for higher categories. For every notion of strict path nn-groupoid P n(X)P_n(X) we get a simplicial nn-category

P n(Y [3])P n(Y [2])P n(Y) \cdots P_n(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} P_n(Y^{[2]}) \stackrel{\to}{\to} P_n(Y)

and there is the need to transmute this into a mere nn-category which is P n(Y)P_n(Y) with lots of “jumps between fibers” thrown in.

I know how to do this for n=2n=2. Konrad and I are writing this up at the moment. And I think I know, operationally, how to do it for any nn.

But I don’t yet quite know the best way to think of the general abstract mechanism at work here.

Posted at March 13, 2008 3:25 PM UTC

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34 Comments & 4 Trackbacks

Simlicial n-categories

From p. 4 of

M. Bullejos, E. Faro and V. Blanco, A full and faithful nerve for 2-categories

I suppose that the answer to my question above will involve the Artin-Mazur codiagonal

W¯:BisimplicialSetsSimplicialSets \bar W : BisimplicialSets \to SimplicialSets

which is right adjoint to the “total decalage” functor

SimplicialSetsBisimplicialSets SimplicialSets \to BisimplicialSets

obtained by pullback along the ordinal sum

+:Δ×ΔΔ. + : \Delta \times \Delta \to \Delta \,.

I need to see if that really yields the result I am looking for:

for instance, is

W¯(P 1(Y [3])P 1(Y [2])P 1(Y)) \bar W \left( \cdots P_1\left(Y^{[3]}\right) \stackrel{\stackrel{\to}{\to}}{\to} P_1\left(Y^{[2]}\right) \stackrel{\to}{\to} P_1\left(Y\right) \right)

equal to the nerve of the the groupoid P 1 Y(X)P_1^Y(X) of definition 2.11?

Posted by: Urs Schreiber on March 14, 2008 4:51 PM | Permalink | Reply to this

all the way

Suppose that about the codiagonal is right. Looks good to me.

Then it’s finally getting pretty close to the full \infty-version of what I am talking about for so long now:

Proposed definition: nonabelian differential \infty-cocycle aka \infty-bundle with connection

a) a smooth space is a sheaf on the category of open subsets of Euclidean space and smooth maps between these;

b) a Lie \infty-groupoid is a Kan complex internal to smooth spaces;

c) for XX a smooth space, write Π (X) \Pi_\infty(X) for the Lie \infty-groupoid whose underlying simplicial smooth space is just the simplicial smooth space S (X)S^\bullet(X) of smooth singular simplices in XX, S n(X)=Hom smoothSpaces(Δ n,X)S^n(X) = Hom_{smoothSpaces}(\Delta^n,X);

d) for π:YX \pi : Y \to X a regular epimorphism of smooth spaces, write

Y Y^\bullet

for the Čech Lie \infty-groupoid whose underlying simplicial smooth space is (Y [3]Y [2]Y) \left( \cdots Y^{[3]} \stackrel{\stackrel{\to}{\to}}{\to} Y^{[2]} \stackrel{\to}{\to} Y \right)

and write

Π Y(X) \Pi_\infty^Y(X)

for the Lie \infty-groupoid whose underlying simplicial smooth space is W¯(Π (Y ))=W¯(Π (Y [3])Π (Y [2])Π (Y)) \bar W\left( \Pi_\infty(Y^\bullet) \right) = \bar W \left( \cdots \Pi_\infty(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} \Pi_\infty(Y^{[2]}) \stackrel{\to}{\to} \Pi_\infty(Y) \right)

e) assume it is right that this comes, as it should, with canonical morphisms

Y Π Y(X)>Π (X) Y^\bullet \hookrightarrow \Pi^Y_\infty(X) \stackrel{\simeq}{\to}\gt \Pi_\infty(X)

f) for gg any finite dimensional L L_\infty-algebra, write S(W(g)) S(\mathrm{W}(g)) for the classifying smooth space for gg-valued forms and S(CE(g)) S(\mathrm{CE}(g)) for the classifying smooth space for flat gg-valued forms;

g) a differential gg-cocycle on a smooth space XX is a choice of regular epimorphism π:YX\pi : Y \to X together with a choice of the horizontal morphisms in the diagram

Y g Π (S(CE(g))) cocycle Π Y(X) (g,A,F A) Π (S(W(g))) connectiondata Π (X) {P i(F A)} Π (S(W(g) basic)) characteristicforms \array{ Y^\bullet &\stackrel{g}{\to}& \Pi_\infty(S(CE(g))) & cocycle \\ \downarrow && \downarrow \\ \Pi_\infty^Y(X) &\stackrel{(g,A,F_A)}{\to}& \Pi_\infty(S(\mathrm{W}(g))) & connection data \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{\{P_i(F_A)\}}{\to}& \Pi_\infty(S(\mathrm{W}(g)_{basic})) & characteristic forms }

in Lie \infty-groupoids;

f) write B nU(1) \mathbf{B}^{n} U(1) for the Lie \infty-groupoid whose underlying simplicial smooth space is the nerve of the strict globular nn-groupoid which is trivial except in degree nn, where it looks like U(1)U(1);

and for BG\mathbf{B} G any one-object Lie \infty-groupoid, write

BBG:= iB n iU(1) \mathbf{B} \mathbf{B} G := \prod_i \mathbf{B}^{n_i} U(1)

for n in_i the degree of the ii-th non-trivial rational cohomology group of the realization BG:=|BG|B G := |\mathbf{B} G|,

write

EG \mathbf{E} G

for the Lie \infty-groupoid whose underlying simplicial smooth space is the image under the décalage functor of BG\mathbf{B} G; this secretly carries a Lie \infty-group structure and we write

BEG \mathbf{B} \mathbf{E} G

for the corresponding one-object Lie \infty-groupoid;

then, for GG any Lie \infty-group, a differential GG-cocycle on a smooth space XX is a choice of regular epimorphism π:YX\pi : Y \to X together with a choice of the horizontal morphisms in the diagram

Y g BG cocycle Π Y(X) (g,A,F A) BEG connectiondata Π (X) {P i(F A)} BBG characteristicforms \array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} G & cocycle \\ \downarrow && \downarrow \\ \Pi_\infty^Y(X) &\stackrel{(g,A,F_A)}{\to}& \mathbf{B}\mathbf{E}G & connection data \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{\{P_i(F_A)\}}{\to}& \mathbf{B} \mathbf{B} G & characteristic forms }

in Lie \infty-groupoids .

Posted by: Urs Schreiber on March 14, 2008 5:47 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Urs,

You may find that the discussion of the codiagonal and its link with homotopy colimits, stacks, the W construction of simplicial fibre bundles and similar things in Richard Lewis’ at

http://www.informatics.bangor.ac.uk/public/math/research/pgpast.html

may help. He has a nice way of thinking of the simplices which is related to one I have seen recently on some link from this blog. (I can’t recall exactly where but you may identify it more quickly than me.)

I do not think that W¯\overline{W} is a good notation for the codiagonal as that has a slightly different, if related, meaning in simplicial theory, and the codiagonal had already been used quite extensively in lots of places before and the usual notation was \nabla. (Balancing Δ\Delta for the diagonal?)

Posted by: Tim Porter on March 16, 2008 1:36 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Thanks!

The explicit description of :BiSsetSset\nabla : BiSset \to Sset recalled on p. 31 is helpful.

It looks pretty plausible that indeed the thing I described is equivalent to

(Π n(Y )) \nabla (\Pi_n(Y^\bullet))

but it also looks not quite trivial to prove.

Maybe as a consitency check:

is it known if

Π (Y )Π (X), \nabla \Pi_\infty(Y^\bullet) \simeq \Pi_\infty(X) \,,

where Π \Pi_\infty simply denotes the simplicial set of (smooth) singular simplices?

Ought to be true. Is it known?

Posted by: Urs Schreiber on March 17, 2008 9:59 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

This sort of thing may be `morally’ true but I have not seen a proof.

I used the codiagonal with Cordier in trying to produce a means of handling homotopy coherent ends and coends. There is also some work recently published with P.J.Ehlers: Ordinal subdivision and special pasting in quasicategories, Adv. in Math. 217 (2007), No 2. pp 489-518; DOI information: 10.1016 / j.aim.2007.05.023 (Preprint available as 05.03 on Bangor preprint server) This shows the relationship between the codiagonal and the ordinal sum of simplices. That is also related to the join operation. Phil Ehlers’ thesis also contained a construction of a simplicially enriched groupoid directly from a space, (only in outline to be honest) but he left mathematics before following that up.

Posted by: Tim Porter on March 17, 2008 2:51 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Thanks again for the information!

I am travelling right now (am in Paris) and have only irregular internet access. Will have to look at the sources you provided later.

But maybe you can quickly give me a hint on this:

suppose we have a morphism

f:S 1S 2 f : S_1 \to S_2

of two simplicial sets and wanted to form its “weak cokernel” i.e. the “homotopy quotient” of S 2S_2 by f(S 1)f(S_1).

It would seem plausible that we can trivially extend ff to a bisimplicial set, which I’ll also denote simply

(S 1fS 2) (S_1 \stackrel{f}{\to} S_2)

and then form the codiagonal of that

(S 1fS 2). \nabla (S_1 \stackrel{f}{\to} S_2) \,.

Could it be that the simplicial set obtained this way plays the role of the desired homotopy quotient?

In particular, if S=BGS = \mathbf{B} G is a simplicial set with a single 0-simplex, I’d like to know if

(BGIdBG) \nabla (\mathbf{B}G \stackrel{\mathrm{Id}}{\to} \mathbf{B}G)

is possibly BEG\mathbf{B} \mathbf{E} G.

Can you say anything about this?

Posted by: Urs Schreiber on March 18, 2008 9:38 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

I am travelling right now (am in Paris)

Heh, what’s the latest news with the Bogdanov brothers? I see they’ve just released a new book there.

Posted by: Bruce Bartlett on March 18, 2008 3:30 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Dunno, am not following the tabloids.

Posted by: Urs Schreiber on March 18, 2008 3:34 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

One of the usual descriptions of homotopy colimits or more generally homotopy coends uses a diagonal of a bisimplicial set. If S 1S_1 and S 2S_2 are Kan I would expect there to be a homotopy equivalence between the thing you write and that description. (There is work by Cegarra and Remedios(Cegarra, A.M., Remedios, J.: The relationship between the diagonal and the bar constructions on a bisimplicial set. Topol. Appl. 153, 21–51 (2005) see also Manuscripta Math. 124 ( 2007), pp. 427–457.) that should imply they are.)

I will try to describe the bisimplicial set in another entry later as I have to do a bit of rough work to get it right - or to be reasonably sure I have it right.

Tim

Posted by: Tim Porter on March 18, 2008 3:56 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

One of the usual descriptions of homotopy colimits or more generally homotopy coends uses a diagonal of a bisimplicial set. If S 1S_1 and S 2S_2 are Kan

Yes, let’s assume that throughout!

I would expect there to be a homotopy equivalence between the thing you write and that description.

That’s very helpful, indeed. Thanks. I am glad we are talking about this.

I will try to describe the bisimplicial set in another entry later

Nice, thanks, I appreciate it! Looking forward to it. Meanwhile I try to follow the references you provided.

Posted by: Urs Schreiber on March 18, 2008 4:02 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

I am a bit confused. My point of confusion is your line `we can trivially extend ff to a simplicial set. What precisely do you mean?

The construction you call homotopy cokernel is what I call the mapping cone of ff. For others who may not know what this looks like, take a cone on S 1S_1 by first taking the cylinder S 1×Δ[1]S_1\times \Delta[1] and then form the pushout of CS 1S 1fS 2CS_1\leftarrow S_1\stackrel{f}{\to} S_2. This is also the homotopy colimit of Δ[0]S 1fS 2\Delta[0]\leftarrow S_1\stackrel{f}{\to} S_2.

Any homotopy colimit has a description as a diagonal of a bisimplicial set, but in this situation the second direction would seem to be almost trivial.

Let me explore this link a bit. (I have found a copy of my last student’s PhD thesis on my laptop and it contains a good description that I will lift.)

We have a functor F:𝕀SSetF : \mathbb{I}\to SSet (so in this case 𝕀\mathbb{I} is the fork category 2012\leftarrow 0 \rightarrow 1). Define the simplicial replacement *F\coprod_* F to be the bisimplicial set with ( *F) p,q:= (I 0f 1f pI p)Ner(𝕀)(FI 0) q(\coprod_*F)_{p,q} := \coprod _{(I_0\stackrel{f_1}{\to} \ldots \stackrel{f_p}{\to} I_p)\in Ner(\mathbb{I})}(FI_0)_q

The bisimplicial set structure is reasonably easy to work out but is given in Richard Lewis’ thesis if needs be.

Then hocolimFdiag *F.hocolim F \simeq diag \coprod_* F.

An additional comment (observed by Richard) is that if GG is an ordinary group acting on a simplicial set YY, think of it as a functor ρ:GSSet\rho :G\to SSet, then W(G)W(G) is the same as hocolimρhocolim \rho. I do not know if that helps with your last question.

Richard’s thesis is a mine of good stuff on both the codiagonal and homotopy colimits of relevance to stacks and things. Some of this is quite well known (e.g. to Jack Duskin), but this is a convenient place to find it gathered together. (It can be found at
www.informatics.bangor.ac.uk/public/math/research/pgpast.html)

The mapping cone forms part of the Puppe sequence of the map ff, and some of what you want for this last question is in Larry Breen’s paper on Bitorsors towards the end. If you cannot find it or want a summary, I can give it, but it would be a bit long perhaps for the blog so I could prepare a short article if that would help. I have some stuff already typed up so can adapt it. It is all `well known’ even `classical’ but that does not make it self evident to people from other backgrounds.

Posted by: Tim Porter on March 18, 2008 5:25 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

My point of confusion is your line ‘we can trivially extend ff to a simplicial set.

Okay, I wrote “It would seem plausible that we can”, but maybe not. Sorry, I was busy all day with something else, will be able to think about this seriously when I am back in my office tomorrow evening. Thanks for bearing with me.

The construction you call homotopy cokernel is what I call the mapping cone of ff

Good, that’s exactly what I am after. I know (from my work with David Roberts) that the BEG\mathbf{B}\mathbf{E} G that I need is the mapping cone of BGIdBG \mathbf{B}G \stackrel{\mathrm{Id}}{\to} \mathbf{B}G but I was trying to see how this could be expressed using the codiagonal, in order to understand the structural similarities between Y (Π (Y))Π (x) Y^\bullet \to \nabla(\Pi_\infty(Y)) \to \Pi_\infty(x) and BGBEGBBG \mathbf{B} G \to \mathbf{B} \mathbf{E} G \to \mathbf{B} \mathbf{B} G (where the last item is uspposed to denote that “rational approximation” BBG=B iB n iU(1)\mathbf{B} \mathbf{B} G = \mathbf{B}\prod_i \mathbf{B}^{n_i} U(1)).

But possibly that particular attempt is mislead.

I could prepare a short article if that would help. I have some stuff already typed up so can adapt it.

I’d certainly be interested in having a look! We can make it a guest post if you like. I’d enjoy it.

And thanks for all the pointers Richard’s work. I’ll have a look.

(Did talk briefly again with Larry Breen yesterday in Paris after my talk, but I’ll have another look at the end of his bitorsor article, as you suggest.)

Posted by: Urs Schreiber on March 18, 2008 10:20 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

With regards to the hocolim (this is also from Lewis’ thesis) we can calculate it up to weak equivalence as hocolimF *F \mathrm{hocolim} F \sim \nabla \coprod_* F with F:CsSetF:C \to sSet.

For the case of interest, the mapping cone, ObjC={0,1,2}Obj C = \{0,1,2\}, giving us only three slice categories to calculate, two of which are trivial - see section 3.3.3 of Lewis’ thesis. Urs would probably want to use the codiagonal at the last step.

Posted by: David Roberts on March 19, 2008 2:00 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Thanks, David, I should have indicated the section. If everything is Kan then the weak equivalence would be a homotopy equivalence, but in fact I would hope that the combinatorial nature of the map should make it clear that it is a direct homotopy equivalence anyway. The codiagonal object should be very easy to calculate using Richard’s notation. (That was one of the very pleasing things about his thesis, that he was very good at the exposition of the known bits as well as the new material.)

I am sure somewhere there is a construction of the mapping cone via the codiagonal but I cannot think where. It should be then easy to compare that form with the more usual one.

There may be some point in mentioning Conduché’s work here. He looked at Loday’s idea of a mapping cone of a map of crossed modules as being a way of going from a crossed square to a 2-crossed module. That seems to be very closely related to your stuff but of course you are in the smooth context as well.

Posted by: Tim Porter on March 19, 2008 8:00 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

That seems to be very closely related to your stuff

Correct! This is a major part of our article - the universal G (2)G_{(2)} “space” (G (2)G_{(2)} a strict 2-group given by a crossed module HGH \to G) is the Gray-group (==semistrict 3-group) coming from the mapping cone of id HGid_{H \to G}, which also coincides with a Gray-group of inner automorphisms of G (2)G_{(2)} on itself.

This has links to the theory of simplicial groups as you will well know, but simplicial groups still have strict multiplication, and what Urs is trying to do is consider objects with weakly coherent multiplication. I suggested in private conversation Kan complexes (topological, if desired) which are A A_\infty-algebras in some fashion, but I do not know if these have been studied before. Perhaps a silent reader can aid me (cough-jim-cough ;).

Posted by: David Roberts on March 19, 2008 8:29 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

I’m not all that silent
Just didn’t see any use for my less than 2 cents (tuppence) worth
but now that you raise the issue of simplicial A_\infty, what are you asking?

If simplicial A_\infty is automaticallyt kan
as simplical group is?

jim (hopeing this is the jim you intended)

Posted by: jim stasheff on March 19, 2008 8:55 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Urs and I differ slightly on what we would consider a weak \infty-group GG. Urs is taking the viewpoint that we know the thing which should act like the classifying space (called BG\mathbf{B}G, with only one 0-simplex), and then we expect the space EG:=Dec 1BG\mathbf{E}G := Dec^1\mathbf{B}G to be the universal GG-bundle, and further that this is itself a weak \infty-group. I can do this in the case that GG is a(n internal) simplicial group, hence a strict \infty-group, but it relies on the algebraic structure coming from GG itself, not just on BG\mathbf{B}G, which is W¯G\overline{W}G in this case.

The underlying simplial set/space of GG in the weak case is the obvious fibre FF of EGBG\mathbf{E}G \to \mathbf{B}G over the only 0-simplex of BG\mathbf{B}G. This is obvious in the case when BG=W¯G\mathbf{B}G = \overline{W}G, but the group structure isn’t. The only thing I know of is Kan’s loop functor (forming an adjoint pair with W¯\overline{W})), but this isn’t the sort of thing one can internalize, seeing as it uses free groups.

FF is formally like a loop space, so I would expect some sort of A A_\infty structure hanging around. So I guess by question is, what has been done on simplicial ‘loop spaces’. I suppose I’m just interested in pointers to A A_\infty-spaces where spaces are actually simplicial sets. I think we have to have Kan from the beginning, as we assume BG\mathbf{B}G is Kan.

Also, I vaguely recall some result about A A_\infty-spaces being homotopy/weakly equivalent to actual topological groups, but no details come to mind.

Posted by: David Roberts on March 20, 2008 5:18 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Urs is taking the viewpoint that we know the thing which should act like the classifying space (called BG\mathbf{B} G, with only one 0-simplex), and then we expect the space EG:=Dec 1BG\mathbf{E}G := Dec^1 \mathbf{B} G to be the universal GG-bundle, and further that this is itself a weak \infty-group.

Right, in what I want to do I don’t really ever need the \infty-group GG itself, but only its incarnation as a one-object \infty-groupoid BG\mathbf{B} G.

And same actually for EG:=Dec 1BG\mathbf{E}G := Dec^1 \mathbf{B} G, which secretly carries a group structure itself – and realizing that group structure on EG\mathbf{E}G should, as we know for low nn, be directly related to realizing the group structure on the fiber of EGBG\mathbf{E} G \to \mathbf{B} G, which you are addressing.

So I adopted the point of view that what I really need is

BEG:=Cone(BGIdBG). \mathbf{B} \mathbf{E} G := Cone(\mathbf{B} G \stackrel{Id}{\to} \mathbf{B} G) \,.

That in a way leads to the reverse problem to the one you emphasize: with that approach one gets fiber and total space for free

BGBEG?? \mathbf{B} G \to \mathbf{B} \mathbf{E} G \to ??

and the problem is to determine what sits on the right, which morally should be “BBG \mathbf{B}\mathbf{B} G ”, interpreted suitably.

This is probably the same problem in disguise, because again it is the question for a secret group structure, this time on BG\mathbf{B} G.

One advantage is that we know that and how this works when GG is sufficiently abelian (sufficiently stable, as John would say): then BBG\mathbf{B}\mathbf{B} G simply exists on the nose, and we indeed get the universal GG-bundle in its groupoid incarnation

BGBEGBBG. \mathbf{B} G \hookrightarrow \mathbf{B} \mathbf{E} G \to \mathbf{B}\mathbf{B} G \,.

So my question is in a way the “co”-version of your question:

you are asking for the fiber of

EGBG, \mathbf{E} G \to \mathbf{B} G \,,

while I am asking for the co-fiber of

BGBEG, \mathbf{B} G \to \mathbf{B}\mathbf{E}G \,,

Both question should actually, I think, be part of one long sequence, which looks like

GEGBGBEGBBG G \to \mathbf{E} G \to \mathbf{B}G \to \mathbf{B} \mathbf{E} G \to \mathbf{B}\mathbf{B} G \to \cdots

along the lines that Mathieu Dupont indicated here in the entry on On Weak Cokernels for 2-Groups.

From slide 616 on I have a movie which is supposed to indicate how that long sequence fits into the general theory of GG-bundles with connection.

Posted by: Urs Schreiber on March 20, 2008 2:11 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Suitably interpreted, that long exact sequence of spaces implies G is an infinite loop space aka E_\infty space
e.g. G abelian or G = the infinite matrix groups O or U or Sp …

Posted by: jim stasheff on March 20, 2008 8:54 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

I am following the

e.g. GG abelian

but how does

GG = the infinite matrix groups OO or UU or SpSp

work?

Is there a

BBU \mathbf{B}\mathbf{B} U

other than that rational approximation

B nB 2nU(1) \mathbf{B}\prod_{n \in \mathbb{N}} \mathbf{B}^{2n} U(1)

??

Posted by: Urs Schreiber on March 20, 2008 9:01 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

There has been some discussion of the use of long homotopy exact sequences (Puppe sequences etc.) and in particular their use for maps between classifying spaces. this was nicely treated in Breen’s Bitorseurs paper.

I have written up a version of some of this, with a few more details in some notes and have put this on my homepage. A direct link to the file is given from the section:
crossed and simplicial algebraic homotopy and non-Abelian cohomology,
of the list of my research interests.

The direct link to the file itself is:

Crossed Menagerie.

(I will post this an hope it works! It has crashed 3 times already when I have tried to check it from the preview.)

This file is the first 7 chapters of a set of notes I prepared for a meeting in Buenos Aires in Dec. 2006, and was extended for use with an MSc class in Ottawa last summer. There may be references to later material than I have included as there are draft versions of later chapters in the process of being typed. I hope they are useful. There is a lot of other stuff on crossed things in there as well.

Posted by: Tim Porter on March 21, 2008 12:23 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Thanks, Tim.
Noohi was here last week and crossed modules came up in his talk. Gerstenhaber had some interesting comments on how they came up - without the name, I think - in his work and other in basic homological algebra.

Posted by: jim stasheff on March 21, 2008 12:50 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

That last is easy. If X is Ainfty, then there exists BX and X is homotopy equivalent to Omega BX (Stasheff) and for reasonable X, e.g. homotopy type of CW, we have Omega BX homotopy equivalent to a top group (Milnor).
For the rest of your questions, why not post the query to the ALG-TOP list?

Posted by: jim stasheff on March 22, 2008 7:12 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Here is another question on the codiagonal:

while it’s an elegant construction in the simplicial context, I still have a bit left of the desire to work with \infty-groupoids with specified composition (meant as opposed to mere Kan complexes with their guarantee of existence of a composition).

For my work I have been reading Street’s Categorical and combinatorial aspects of descent for a long time like a prophecy: a text full of deep meaning which only eveals itself after lots of contemplation and inspiration.

Street discusses “nonabelian” cohomology for several kinds of weakened \infty-categories, but most attention is paid to strict ω\omega-categories.

But then of course, what appears prominently are cosimplicial ω\omega-categories. These are what we obtain a descent ω\omega-category and a cohomology theory from.

For the example I mentioned we’d have that (smooth) simplicial space

Y =(Y [2]Y) Y^\bullet = (\cdots Y^{[2]} \stackrel{\to}{\to} Y )

coming from a regular epimorphism YX Y \to X of (smooth) spaces;

and with a choice of (Lie) ω\omega-group BG\mathbf{B} G would form the cosimplicial ω\omega-category

E:=Hom(,BG)(Y ), E := Hom(--, \mathbf{B} G)(Y^\bullet) \,,

where in the first entry we regard each space as the disrete ω\omega-category over it.

The descent ω\omega-category DescE\mathrm{Desc} E of that is the “nonabelian cochain” category of XX with values in GG and its decategorification is the nonabelian cohomology of XX with values in BG\mathbf{B} G.

As you know, what me and collaborators are looking at is refining this to what should be called differential nonabelian cohomology, essentially by using instead the cosimplicial category

E¯:=Hom(Π (),BG)(Y ). \bar E := Hom(\Pi_\infty(--), \mathbf{B} G)(Y^\bullet) \,.

But that’s not the full story. As discussed in the entry above, the full story requires that I look at something like the codiagonal of the simplicial ω\omega-category Π (Y ). \Pi_\infty(Y^\bullet) \,. As we have discussed, I can take this at the level of bisimplicial sets after using the ω\omega-nerve to send

ωCatSSet. \omega Cat \to SSet \,.

But, given the setup, I’d much rather form a “codiagonal” of Π (Y )\Pi_\infty(Y^\bullet) which is not any old simplicial set, but again an ω\omega-category.

Does such a codiagonal

:simplicialωcategoriesωcategories \nabla : simplicial \omega-categories \to \omega-categories

exist?

Alternatively:

Does a codiagonal

:simplicial(ωCat)categories(ωCat)categories \nabla : simplicial (\omega-Cat)-categories \to (\omega-Cat)-categories

exist?

Posted by: Urs Schreiber on March 19, 2008 9:52 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Something akin to the codiagonal exists I think. In the paper I mention (Adv in Maths end of last year) I used old results of Phil Ehlers and even older ones of Cordier and myself to explore pasting in quasicategories. These results were taken up by Dominic Verity and interpreted for his complicial sets, and there you are getting very near what you want, I believe. It depends on your favorite flavour of ω\omega-categories perhaps.

You would need tensors over simplicial sets or similar for the construction to be adaptable to the ω\omega-cat. context. That should work if you have coproducts.

There is a coend formula for the codiagonal on bisimplicial sets, but it does not trivially extend to one for simplicial ω\omega-categories:

if XX is a bisimplicial set, (X) n= [p],[q]Δ([p]+[q],[n])×X p,q(\nabla X)_n = \int^{[p],[q]}\Delta([p]+[q],[n])\times X_{p,q}where + is ordinal sum. I suspect something a bit subtler may be needed if XX is simplicial in pp but ω\omega-categorical in qq. If a simplicial model of ω\omega-categories is used (quasicat or complicial) then probably its ok. I think Verity handles the latter situation although probably not explicitly.

Posted by: Tim Porter on March 19, 2008 11:09 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

One thought and a question: any simplicial group has a choice of `specifiable composition’ as the filler condition for a Kan complex structure is given constructively by an algorithm. (For a simplicially enriched groupoid, I do not know if a similar algorithm has been written down in each hom-set.)

If you are doing things simplicially, and geometrically at the same time, then there is an algebraic composition given by these fillers and also a geometric one via dividing out by some sort of thin homotopy relation (although that is sometimes much harder to pin down than is first thought as you well know). The two are not the same and that is a pain!

My question is how do you handle the difference? The answer is probably in the archives of this blog or your papers (which as usual I have printed out but not yet read!), but some brief intuitive idea would suffice.

When Ronnie, Jim Glazebrook and I were trying to define a smooth crossed complex for a smooth space, we considered using the quotient of the simplicially enriched groupoid by thin elements, defined algebraically via the filling algorithm. That seemed to work but for various reasons of a non-mathematical nature we never wrote it all down and so details have not been tried and tested as they should be.

Posted by: Tim Porter on March 19, 2008 1:42 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

any simplicial group has a choice of ‘specifiable composition’ as the filler condition for a Kan complex structure is given constructively by an algorithm.

While I didn’t know that, it sounds plausible: I am thinking of simplicial groups as essentially strict \infty-groups. That’s certainly what they are for low nn.

And so I am actually not so much interested in having the theory for simplicial groups. Rather, if we talk simplicially at all, I’d want to consider one-object Kan complexes as (classifying groupoids of) \infty-groups.

how do you handle the difference? The answer is probably in the archives of this blog or your papers

Wait, there must be some misunderstanding here. It doesn’t seem to me that what you just considered is something I ever talked about before. In fact, I am not entirely sure I understand precisely which problem you have in mind running into when attemtping to

define a smooth crossed complex for a smooth space

For me a smooth crossed complex is a crossed complex internal to groupoids internal to smooth space (manifolds, Chen-smooth spaces, more general sheaves, some choice).

But I must be missing the point you have in mind. Maybe if you say it again it becomes clear. Sorry for being dense…

Posted by: Urs Schreiber on March 19, 2008 10:40 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Let me try to explain this point by an example. First three facts.

(i) The singular complex of any space is a Kan complex with fillers given `geometrically’ using retractions of the models.

(ii) The underlying simplicial set of a simplicial group is a Kan complex with fillers given ALGEBRAICALLY by the algorithm given in Peter May’s little book and which you can find also on pages 24 and 25 of the Crossed Menagerie notes that I have just posted on the blog.

(iii) If GG is a topological group, its singular complex is a simplicial group (topological if you need it).

We thus have that there are two different formulae for getting fillers in Sing(G)Sing(G) one geometric, one algebraic. for any horn in Sing(G)Sing(G), the difference between the two fillers is quite interesting. (Phil Ehlers did some work on this in his thesis. The difference is closely linked to trying to get ultimate higher van Kampen theorems, but that is another story.)

It is also interesting to see what the different notions of thinness that are natural to the two situations give. In one there is always a retraction based filler, this thus goes through a lower dimension so it is intuitively `thin’. The other natural notion of thinness is as a product of degeneracy elements. Here the algebraic fillers are thin in that sense. As the degeneracy operators in Sing(G)Sing(G) are continuous, and multiplication and inversion are as well, the filling maps from the space of all horns of a given type can be seen to be continuous if the algorithm is used.

The \infty-group structure of a simplicial group can as you claim be specified via the Kan complex structure, but I am not certain that it is strict if you use the algorithmic definition of fillers. For instance, the intersection of the Moore complex with the subgroup generated by degenerate elements in each dimension is usually non-trivial. If it is trivial then the Moore complex is a crossed complex and so has vanishing Whitehead products etc.

The formula for Whitehead and Samelson products for a simplicial group is given in terms of products of commutators of degenerate elements so lives in that intersection. (The formula is given without proof in Curtis’ old survey article on Simplicial Homotopy theory. A proof does not seem to be in the literature, but if someone knows of a published version i would appreciate a reference.)

Getting back to strictness, unless I am getting my thoughts all twisted up, associativity of the composition coming from the algebraic Kan fillers is only true if the intersection I mention is trivial in dimension 2, a result that I call the Brown-Loday lemma. The obstruction is related to the Gray tensor product at the categorical level.

I will stop there to await reaction since there may be points that I have misunderstood and others that may not be clear to others.

Posted by: Tim Porter on March 21, 2008 1:03 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Thanks for all the info. But one remark, you write:

The \infty-group structure of a simplicial group can as you claim be specified via the Kan complex structure, but I am not certain that […]

This seems to be the source of the previous misunderstanding: I don’t think I made any claim about any Kan complex structure on simplicial groups.

I think what I said is that if we model \infty-categories by simplicial structures at all, then I would tend to address one-0-simplex Kan simplicial sets as \infty-groups instead of looking at simplicial groups. That remark was more on strategy than on content, because for me part of the goal here is to figure out which models best to use in a certain application I have.

That said, let me enter the question you are addressing after all and ask this question:

if we are to compare the two models of “\infty-groups” that we are discussing,

a) simplicial groups

b) one-0-simplex Kan complexes

and ask how to think of the former as an example of the latter, I would think that the “right” translation operation is not to find a Kan structure on the simplicial set underlying the simplicial group, but rather to use

W¯:SGrpSSet \bar W : SGrp \to SSet

and find a hopefully obvious Kan structure on the result.

But maybe what you describe applies to this?

Posted by: Urs Schreiber on March 21, 2008 7:59 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

As you no doubt know, W¯G\overline{W}G has an explicit Kan structure if GG is the simplicial group of a strict 2-group / crossed module. I would guess that there were explicit fillers to be found in all dimensions in general. I have a vague recollection of seeing this somewhere or writing it down myself, but can only find the case I mention in the literature (in fact in my Yetter TQFT paper, section 3.4). The argument should be to use the inclusion of the horn Λ k[n]\Lambda^k[n] in Δ[n]\Delta[n] map it via the GG-loop group functor, over into simplicially enriched groupoids and analyse the result. That suggests a way of extending with a formula, which being algebraic will translate to the topological case without difficulty.

Posted by: Tim Porter on March 23, 2008 9:47 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

I think that I remember now. The indeterminancy of the lifting in W¯G\overline{W}G is given by the Moore complex terms of GG.

This is essentially due to the calculations in Pilar Carrasco’s thesis where she considers hyper-crossed complexes and the `ultimate Dold-Kan theorem’, but needs the sort of analysis I mentioned on my earlier reply as well.

Her recipe (for GG) involves looking at how a simplicial group is constructed from lower order skeleta using a combination of pairings and extensions. It uses the semidirect decomposition of each dimension to derive the necessary description.

Now one tries to carry that across to W¯G\overline{W}G, the low diemsnional cases give some ideas on how to proceed and guessing a formula and checking it works. Perhaps something along this line can also be constructed from Nick Ashley’s thesis BIBTEX:@Article{ashley,

author = {Ashley, N.},

title = {Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan},

journal = {Dissertations Math.},

year = {1989},

volume = {165},

pages = {11 - 58},
}

Posted by: Tim Porter on March 23, 2008 11:48 AM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

Now that I am back online after a long train ride from Paris to Hamburg, and before replying to Tim Porter’s kind remarks, I’d first want to reply to my own question, since, now that I have thought about it, it is embarrasing that I asked this question in the same comment in which I mentioned Street’s article.

So:

the ω\omega-Category which “coequalizes a simplicial ω\omega-Category” is its codescent object.

For instance for π:YX\pi : Y \to X a regular epimorphism, Y Y^\bullet the simplicial space given by the nerve of π\pi, we have that

codesc(Y ) codesc(Y^\bullet)

is the Čech groupoid of π\pi:

Ner(codesc(Y ))=Y . Ner(codesc(Y^\bullet)) = Y^\bullet \,.

For P:spacesCatP : spaces \to Cat the path groupoid functor, we have

codesc(P(Y ))=P Y(X) codesc(P(Y^\bullet)) = P^Y(X)

the path groupoid of XX relative to YY, that thing which agreed above is naturally to conjecture that it satisfies

Ner(P Y(X))(Ner(P(Y ))) Ner(P^Y(X)) \simeq \nabla( Ner(P(Y^\bullet)))

with \nabla the codiagonal of simplicial sets.

This is in fact actually an example for the theory Street discusses from section 6 on.

More on that later. I think I’ll prepare an entry with an exegesis of Street’s descent theory and its application to the higher differential cohomology that I am talking about.

Posted by: Urs Schreiber on March 19, 2008 10:06 PM | Permalink | Reply to this

Re: Slides: On Nonabelian Differential Cohomology

I wrote:

I think I’ll prepare an entry with an exegesis of Street’s descent theory and its application to the higher differential cohomology that I am talking about.

While I won’t be able to do this today, I should maybe quickly mention how the things about differential cohomology which I am talking about in the above entry look like in the language that Street develops from p. 11 on:

in the context I am sticking to, Street’s ambient category

C C

is the category of smooth spaces (as in space and quantity).

A typical element in

Hom(C op,ωCat) Hom(C^op, \omega Cat)

which we encounter here is

Π n:C opωCat \Pi_n : C^\op \to \omega Cat

which sends any smooth space to its strict path nn-groupoid.

Notice that ordinary (meaning: non-differential) cohomology of a space XX with values in the \infty-groupoid BG\mathbf{B} G is, in Street’s language

Desc(Hom(Disc(),BG),Y ) Desc( Hom(Disc(--), \mathbf{B}G), Y^\bullet)

for Y Y^\bullet the nerve of a good cover π:YX\pi : Y \to X.

(Here Disc(U)Disc(U) denotes the discrete ω\omega-category over the space UU, a notation that Street suppresses throughout, something one certainly wants to do when everything is clear, but which confused me right from his p. 3 on for quite a while. )

I think that essentially all I ever said (and am writing up with Konrad Waldorf) about smoothly locally ii-trivializable nn-transport is pretty much working out the implications of Street’s theory for taking the “object of values” XX (now as on p. 12) to be

X:=Hom(P n(),BG):C opωCat. X := Hom(P_n(--), \mathbf{B}G) : C^{op} \to \omega Cat \,.

In particular with that choice the equation he hides somewhere on p. 22, where it reads

(Xe)x(Xz)t (X e)x \simeq (X z)t

is precisely the “magic square”

P n(Y) P n(X) BG T \array{ P_n(Y) &\stackrel{}{\to}& P_n(X) \\ \downarrow &\Downarrow^\simeq& \downarrow \\ \mathbf{B} G &\to& T }

whose central importance I kept emphasizing (for instance in First edge of the Cube or in Local Transition of Transport, Anafunctors and Descent of nn-Functors or at many other places).

For the record, with that choice Street’s category Loc(t,e)Loc(t,e) is what was finally called Trans(π,i)Trans(\pi,i) (functors that do admit a local trivialization) and what Street calls Q(t,e)Q(t,e) is what was in the end called Triv(π,i)Triv(\pi,i) (functors with chosen local trivialization) in 0705.0452.

Anyway, this just to briefly indicate how the language I am using matches with Street’s (very) general theory of descent.

So then, here is the main statement about what we want to address as nonabelian differential cohomology, schematically:

For PP a BG\mathbf{B} G descent object and ω\omega a globally defined Hom(Π(),BG)Hom(\Pi(--),\mathbf{B}G) descent object, we can say that a differential refinement of PP by ω\omega is a Hom(Π(),BEG)Hom(\Pi(--),\mathbf{B}\mathbf{E}G) descent object \nabla, together with morphisms

P ω bundle connection charclasses. \array{ P &\to& \nabla &\to& \omega \\ bundle && connection && char classes } \,.

Unwrapping all this yields the diagrams of the kind

Y BG Π Y(X) BEG Π(X) BBG \array{ Y^\bullet &\to & \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi^Y(X) &\to & \mathbf{B} \mathbf{E} G \\ \downarrow && \downarrow \\ \Pi(X) &\to & \mathbf{B} \mathbf{B} G }

that I am talking about above.

But enough. More details later.

Posted by: Urs Schreiber on March 19, 2008 11:21 PM | Permalink | Reply to this
Read the post Crossed Menagerie
Weblog: The n-Category Café
Excerpt: Detailed notes by Tim Porter on "crossed gadgetry and cohomology in algebra and topology".
Tracked: March 21, 2008 8:11 PM
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Weblog: The n-Category Café
Excerpt: A discussion of differential nonabelian cocycles classifying higher bundles with connection in the context of the general theory of descent and cohomology with coefficients in infnity-category valued presheaves as formalized by Ross Street.
Tracked: March 22, 2008 7:51 PM
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 4:44 PM
Read the post Higher Hochschild Cohomology and Differential Forms on Mapping Spaces
Weblog: The n-Category Café
Excerpt: On "brane topology", the higher order generalization of Hochschild cohomology. And on Chen iterated integrals.
Tracked: April 18, 2008 7:55 PM

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