September 22, 2008

Group Cocycles and Simplices

Posted by Urs Schreiber

Christoph Wockel asks me to forward the following question to the $n$-Café:

What is a reference for a generalisation of the following canonical cocycle to higher dimensions?

Let $G$ be a connected (locally contractible) topological group and take a section $a : G \to P G \,,$ continuous on a neighbourhood of the neutral element, of the endpoint evaluation map $ev : P G \to G$ from the space $P G$ of continuous paths starting at the neutral element of $G$. Then the assignment $(g,h) \mapsto [a(g)+g \cdot a(h)-a(g h)]$ is a $\pi_1(G)$-valued group 2-cocycle on $G$, describing the universal cover of $G$ as a central extension. Moreover, this cocycle is universal for discrete groups.

A similar construction works in higher dimensions, yielding for each $(n-1)$-connected group $G$ a $\pi_n(G)$ valued group $(n+1)$-cocycle. Moreover, this cocycle is universal for discrete groups. I’ve been searching for a reference for this, but did not succeed.

Thanks for any hints.

My remark: notice that the construction this question is about is closely related to the construction of Čech cocycles for characteristic classes by Brylinski and McLaughlin which I talked about here.

Posted at September 22, 2008 9:50 AM UTC

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Re: Group Cocycles and Simplices

You’re saying there’s a god-given $\pi_n(G)$-valued $(n+1)$-cocycle on a sufficiently nice topological group $G$.

Instead of focusing on the specific cocycle, let’s think about its cohomology class. Then you’re saying that if $G$ is sufficiently nice, there’s a god-given element of

$H^{n+1}(G,\pi_n(G))$

This is group cohomology. But this is isomorphic to

$H^{n+1}(K(G,1),\pi_n(G))$

where $K(G,1)$ is the 1st Eilenberg–Mac Lane space of $G$.

But cohomology is represented by Eilenberg–Mac Lane spaces, so the above group is isomorphic to

$[K(G,1),K(\pi_n(G),n+1)]$

where the square brackets mean ‘the set of homotopy classes of maps’.

I would like to keep chewing away on this until I see that the god-given element you’re talking about is something like an identity map in disguise! But I seem to be stuck right now.

If we take advantage of your assumption that the topological group $G$ is $(n-1)$-connected, then we can use the Hurewicz theorem, which says that in this case

$\pi_n(G) \cong H_n(G)$

I’m not sure how this helps, but I thought I should mention it.

Posted by: John Baez on September 23, 2008 3:55 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

If $G$ is a topological group, what is a $K(G,1)$? Did you mean $BG$?

There are spaces where the fundamental group isn’t discrete (Hawaiian earring, say), but that’s getting a bit much :)

Posted by: David Roberts on September 24, 2008 12:57 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

David wrote:

If $G$ is a topological group, what is a $K(G,1)$? Did you mean $B G$?

No, I meant the classifying space of the underlying group $G_{disc}$ of the topological group $G$:

$K(G,1) := B G_{disc}$

For lurking nonexperts, let me expand on this.

For any sufficiently nice topological group $G$, there is a classifying space $B G$ such that isomorphism classes of principal $G$-bundles over a paracompact space $X$ are in 1-1 correspondence with homotopy classes of maps

$X \to B G$

When we’ve got a discrete topological group $G$ — or in other words, just a plain old group — we usually call the classifying space of $G$ an Eilenberg–Mac Lane space $K(G,1)$. This may alternatively be described as the pointed connected space with $\pi_1 = G$ and all higher $\pi_n$’s trivial.

Any topological group $G$ has an underlying discrete group, say $G_{disc}$. There’s a continuous homomorphism

$G_{disc} \to G$

and by functorial abstract nonsense this gives a map of pointed spaces

$B G_{disc} \to B G$

If we agree that the only sensible meaning of $K(G,1)$ is $B G_{disc}$, then we can say this is a map

$K(G,1) \to B G$

When $G$ is a Lie group, here’s a nice way to think about this. $B G$ is the classifying space for $G$-bundles, while $BG_{disc} = K(G,1)$ is the classifying space for $G$-bundles equipped with a flat connection — or flat $G$-bundles, for short. The map

$K(G,1) \to B G$

comes from the fact that any flat $G$-bundle gives a $G$-bundle.

There are very nice relationships between this stuff and ‘secondary characteristic classes’, like Chern–Simons classes.

I wish I understood $B G_{disc}$ better even for quite simple Lie groups, like $G = \mathrm{U}(1)$ and $G = SU(2)$.

For example: what’s the cohomology of $B G_{disc}$ in these cases? I know some nice elements: secondary characteristic classes. But what’s the whole story?

Posted by: John Baez on September 24, 2008 7:21 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

When $G$ is a Lie group, here’s a nice way to think about this. $B G$ is the classifying space for $G$-bundles, while $B G_{disc} = K(G,1)$ is the classifying space for G-bundles equipped with a flat connection — or flat $G$-bundles, for short.

Incidentally, this is the answer to the question I asked here on January 29 and, after Jim made a remark, again on January 30. Thanks for the reply!! :-)

This was the context of my question back then: there are nice models for the rational approximation of $K(G,1)$ for $G$ any $\infty$-Lie group which are

- (generalized) smooth.

And this is closely related to $\infty$-Lie integration and various other things:

namely, given an $L_\infty$-algebra $g$, there is a generalized smooth space (a sheaf on manifolds) which I keep calling by the weird name $S(CE(g))$ which is the smooth classifying space for flat $g$-valued diffrential form data – or in other words: trivial $G$-$\infty$-bundles with chosen smooth flat $\infty$-connection.

In particular, the fundamental smooth $\infty$-groupoid of these spaces is the one-object $\infty$-groupoid coming from the simply connected smooth $\infty$-group $G$ integrating $g$.

$G = Aut_\bullet(\Pi_\infty(S(CE(g)))) \,.$

This maybe slightly scary chain of symbols is nothing but the analog of

$G = \pi_1(K(G,1))$

for $G$ an ordinary group.

Posted by: Urs Schreiber on September 25, 2008 12:07 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

You’re saying there’s a god-given $\pi_n(G)$-valued $(n+1)$-cocycle on a sufficiently nice topological group G.

Instead of focusing on the specific cocycle, let’s think about its cohomology class. Then you’re saying that if G is sufficiently nice, there’s a god-given element of

(1)$[K(G,1),K(\pi_n(G),n+1)]$

where the square brackets mean ‘the set of homotopy classes of maps’.

Here $G$ is $(n-1)$-connected, so $K(G,1) = BG$ is $n$-connected with $\pi_{n+1}(BG) = \pi_n(G)$. So the first non-trivial stage in the Postnikov tower is

(2)$K(G,1) \rightarrow K(\pi_n(G), n+1),$

exactly the sort of map you are looking for.

Posted by: Dan Christensen on September 24, 2008 3:03 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I wrote:

Here $G$ is $(n−1)$-connected, so $K(G,1)=BG$ is $n$-connected …

I was jumping to the conclusion that when John wrote $K(G,1)$ for a topological group $G$, he meant $BG$, but from a later message of John’s I see that he actually meant to regard $G$ as a discrete group when it appears in the expression $K(G,1)$.

My message gave a map $BG \rightarrow K(\pi_n(G), n+1)$, and John points out that there is a natural map $K(G,1) \rightarrow BG$, so composing these must give the natural map

(1)$K(G,1) \rightarrow K(\pi_n(G), n+1)$

that is being looked for.

Posted by: Dan Christensen on September 26, 2008 1:15 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I should add that I have the vague feeling you’re describing the Postnikov tower of $G$, or something like that.

The Postnikov tower of a connected space $X$ describes this space in terms of the groups $\pi_n(X)$ and certain ‘Postnikov invariants’

$k_n \in H^{n+1}(X_{n-1}, \pi_{n}(X))$

where $X_{n-1}$ is $X$ with its homotopy groups above the $(n-1)$st killed.

The funny pattern of $n$, $n+1$ and $n+2$ in the above formula reminds me of your sentence

… yielding for each $(n−1)$-connected group $G$ a $\pi_n(G)$ valued group $(n+1)$-cocycle.

If you’ve never thought about Postnikov towers, and you like higher categories, there might conceivably be someplace to start learning about them that’s even worse than my lectures on cohomology and $n$-categories.

Posted by: John Baez on September 23, 2008 7:08 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I should add that I have the vague feeling you’re describing the Postnikov tower of $G$, or something like that.

I still haven’t quite found the time to think about this in the detail that I ought to, but it seems that this is a way of looking at the construction of higher String-like extensions by successively killing homtopy groups.

In fact, if I think about it for a bit it feels that I should be able to translate this directly to the construction of the strict String 2-group and of the strict Fivebrane 6-group by $L_\infty$-integration in those notes on Twisted nonabelian differential cohomology that some of you have seen.

One nice aspect of Christoph’s point of view, in contrast to the $L_\infty$-integration picture, is that it allows to talk not only about killing of non-torsion homotopy groups, but also about the torsion homotopy groups.

In particular, while I can express the steps $Spin(n) \mapsto String(n) \mapsto Fivebrane(n)$ in terms of $\infty$-Lie theory, I can’t look at it this way for the first two steps $O(n) \mapsto S O(n) \mapsto Spin(n)$ in the same way, since that requires killing two torsion groups ($\mathbb{Z}_2$). But I gather using Christoph’s point of view this is immediate:

As he points out, his 2-cocycle for $O(n)$ characterizes the extension $\mathbb{Z}_2 \to Spin(n) \to S O(n) \,.$ I suppose if we do a bit of “negative thinking” a la Toby Bartels we can tell a story how even one step further down his construction yields a 1-cocycle which characterizes $S O(n)$ with respect to $O(n)$. Hm…

As Hisham Sati taught me, there is indication that for the purpose of physics we need to go up to the 11-connected cover of $Spin(n)$ given by the strict 10-group to be called $Ninebrane(n)$. The NS-9-brane is the “space filling nine-brane at the Hořava-Witten-end-of-the-world” or the like, and one expects some curious properties, with its worldvolume theory being closely related to the target space theory of the heterotic string.

Hisham has some articles on the arXiv with tentative discussion of this point, and he tells me that some other group recently was able to make some aspects of these NS 9-branes more concrete.

In any case. To get to $Ninebrane(n)$ involves not just killing $\pi_{11}(O(n)) = \mathbb{Z}$ but first also the two torsion groups $\pi_{8}(O(n)) = \pi_{9}(O(n))= \mathbb{Z}_2$.

Posted by: Urs Schreiber on September 23, 2008 10:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Urs wrote:

I still haven’t quite found the time to think about this in the detail that I ought to, but it seems that this is a way of looking at the construction of higher String-like extensions by successively killing homotopy groups.

Let me warn you: there’s a big difference between ‘killing’ homotopy groups and ‘cokilling’ cohomology groups. The first is fundamental to Postnikov towers. The second is fundamental to constructing the string group. I was confused for a long time about the difference!

To ‘kill’ a specific element $\alpha$ of the $n$th homotopy group of a space $X$, we first pick a map from $S^n$ to $X$ representing $\alpha$. Then we glue an $(n+1)$-ball to $X$ using this map. We get a new space $\tilde{X}$ and an inclusion $X \to \tilde{X}$. Pushing forward along this inclusion sends the element $\alpha \in \pi_n(X)$ to $0 \in \pi_n(\tilde{X})$.

To ‘cokill’ a specific element $\beta$ of the $n$th cohomology group of a space $X$, we first pick a map from $X$ to $K(\mathbb{Z},n)$ representing $\beta$. Then we take the homotopy fiber of this map. We get a new space $\tilde{X}$ and a fibration $\tilde{X} \to X$. Pulling back along this inclusion sends the element $\beta \in H^n(X)$ to $0 \in H^n(\tilde{X})$.

These constructions are ‘dual’ in a certain sense. I could make the duality even more evident if I had used phrases like ‘homotopy cofiber’ and ‘cofibration’ when describing how to kill homotopy groups, to match my use of ‘homotopy fiber’ and ‘fibration’ when describing how to cokill cohomology groups. But that would make the simple process of gluing on a ball seem more scary!

To form the topological group $String(G)$, we take our compact simply-connected simple Lie group $G$ and cokill the generator of its 3rd cohomology group. This has the side-effect of making its 3rd homotopy group trivial, but it’s not the same as killing the generator of the 3rd homotopy group. You can sense this by noting that we have a nice map $String(G) \to G$, not $G \to String(G)$.

I think some people say $String(G)$ is built by ‘killing the 3rd homotopy group’ of $G$. I used to be guilty of this myself! But this is a sloppy usage, different from what homotopy theorists mean when they speak of ‘killing homotopy groups’ in the subject of Postnikov towers. So, somewhere around the time we wrote our paper on the string group, I started talking about ‘taking the homotopy fiber of the map $G \to K(\mathbb{Z},3)$ that represents the generator of the 3rd cohomology’. The point is not to sound more fancy: it’s to distinguish two different constructions with very different properties!

Posted by: John Baez on September 24, 2008 1:52 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Some terminology, to help people who want to look up these things in, say, Hatcher’s freely available book.

The “cokilling” construction is called forming the $n$-connected cover, since it generalizes the universal cover.

And the resulting tower of spaces is usually called the Whitehead tower.

Posted by: Dan Christensen on September 24, 2008 3:08 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Thanks! I knew the ‘$n$-connected cover’ terminology, but forgot to mention it here. I hadn’t heard the term ‘Whitehead tower’.

Long time no see!

Posted by: John Baez on September 24, 2008 7:03 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Whitehead towers were mentioned in the thread on 2-covers. I see I was guilty of the killing/co-killing confusion.

Posted by: David Corfield on September 24, 2008 8:45 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Whitehead tower in the west and, dually, Postnikov tower in Moscow, were discovered about in the same period.
I think that I read in the volume Golden Year of Moscow Mathematics (Smilka Zdravkovska ed.) or elsewhere
about Whitehead being in Moscow visiting about the same time, but somehow there happened that Postnikov and Whitehead did not realized at the time that they were
working on a variant of the same idea. If somebody can check out the exact story…I do not have the book.

There are some other interesting historical points, e.g. the role of Pontrjagin, Aleksandrov and Kurosh in advising Postnikov what to do with his discovery. Pontrjagin was Postnikov’s advisor, otherwise helpful, but his advice was not crucial in estimating the importance of the construction, Aleksandrov did not really help, and Kurosh who was not a topologist but asked a right question.

Namely Postnikov devised his method for a particular computation, and Kurosh was asking about what else you can calculate with this, forcing the conversation to the point where Postnikov realized that knowing all “Postnikov invariants” in addition to the homotopy groups he can calculate EVERYTHING, hence that the whole set in fact determines the homotopy type.

Posted by: Zoran Skoda on October 7, 2008 5:44 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Let me warn you: there’s a big difference between ‘killing’ homotopy groups and ‘cokilling’ cohomology groups. The first is fundamental to Postnikov towers. The second is fundamental to constructing the string group.

Thanks for the correction! (Or is it a rrection?)

I think some people say $String(G)$ is built by ‘killing the 3rd homotopy group’ of $G$.

Yes. Not just some, it seems. Okay, better late than never… Thanks again.

Posted by: Urs Schreiber on September 25, 2008 11:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Is there a reason for the lack of correspondence between:

killing homotopy groups

and

cokilling cohomology groups?

You might have expected

cokilling cohomotopy groups,

but I guess cohomotopy has been given a different sense.

Posted by: David Corfield on September 25, 2008 12:16 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Is there a reason for the lack of correspondence between:

killing homotopy groups

and

cokilling cohomology groups?

You might have expected

cokilling cohomotopy groups,

The Eckmann-Hilton dual of homotopy is cohomology, just as the Eckmann-Hilton dual of a sphere is a $K(Z,n)$.

Up to (weak) homotopy, all spaces are built from spheres, and up to (weak) homotopy, all spaces are co-built from Eilenberg-Mac Lane spaces.

Eckmann-Hilton duality is discussed in Hatcher’s book, 4.H.

Posted by: Dan Christensen on September 26, 2008 12:43 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Thanks. Hatcher writes

There is a somewhat deeper duality between homotopy groups and cohomology, which one can see in the fact that cohomology groups are homotopy classes of maps into a space with a single nonzero homotopy group, while homotopy groups are homotopy classes of maps from a space with a single nonzero cohomology group. (p. 462).

The heuristic aspect to it is interesting,

Eckmann–Hilton duality can be extremely helpful as an organizational principle, reducing significantly what one has to remember, and providing valuable hints on how to proceed in various situations. To illustrate, let us consider what would happen if we dualized the notion of a Postnikov tower of principal fibrations, where a space is represented as an inverse limit of a sequence of fibers of maps to Eilenberg–MacLane spaces. In the dual representation, a space would be realized as a direct limit of a sequence of cofibers of maps from Moore spaces.

This entry notes that Eckmann-Hilton duality is

A duality principle variously described as “…a metamathematical principle that corresponding to a theorem there is a dual theorem (each of these dual theorems being proved separately)” [a4], “…a guiding principle to the homotopical foundations of algebraic topology…” [a1], “…a principle or yoga rather than a theorem” [a5], and “…a commonplace of experience among topologists, accepted as obvious” [a3]. The duality provides a categorical point of view for clarifying and unifying various aspects of pointed homotopy theory, but is often heuristic rather than strictly categorical.

Is it known when this strictness breaks down so that

It is not true, however, that dual theorems necessarily admit dual proofs,

and

It is also possible that the dual of a theorem is false?

Posted by: David Corfield on September 26, 2008 8:59 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Oh, my question over there under Math Miniatures
is answered here. What a tangled web we weave!

Posted by: jim stasheff on September 27, 2008 1:48 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Viro and Fuchs display the duality in two columns on pp. 17-20 of Topology II.

Posted by: David Corfield on September 26, 2008 9:49 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

David wrote:

Is there a reason for the lack of correspondence between

killing homotopy groups

and

cokilling cohomology groups?

There’s actually a perfect correspondence in everything except the terminology. Dan Christensen explained it, but perhaps a bit too tersely for the nonexpert. So:

The $n$th homotopy group classifies maps from $S^n$ into a space, where $S^n$ is the space whose only interesting cohomology group is the $n$th one, which is $\mathbb{Z}$.

The $n$th cohomology group classifies maps from a space into $K(\mathbb{Z},n)$, where $K(\mathbb{Z},n)$ is the space whose only interesting homotopy group is the $n$th one, which is $\mathbb{Z}$.

Note that thanks to these two sentences, it follows that

$K(\mathbb{Z},n)$ is the space whose only interesting homotopy group is the $n$th one, which is $\mathbb{Z}$.”

means the exact same thing as

$S^n$ is the space whose only interesting cohomology group is the $n$th one, which is $\mathbb{Z}$.”

So, everything is perfect under heaven.

You might have expected

cokilling cohomotopy groups,

Or you might have expected “killing homology groups”. But those things both play other roles.

Posted by: John Baez on October 7, 2008 8:51 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Are people interested in

$X$ is the space whose only interesting cohomotopy group is the $n$th one,

or

$X$ is the space whose only interesting homology group is the $n$th one?

Posted by: David Corfield on October 8, 2008 8:49 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Above David was trying to understand why there is much ado about

$cohomology \leftrightarrow homotopy$

while at the same time nobody seems to care much about cohomotopy , much less about any duality of that to homology.

Experts will correct me, but I have recently come to think that the reason may be this:

cohomology and homotopy are both fundamental concepts, as follows: cohomology is about gluing and descent of presheaves, hence about $\infty$-stacks, whereas homotopy is about codescent of co-presheaves, hence $\infty$-costacks.

I have talked about that recently here.

Taking this as fundamental, homology in turn seems to be a rather contrived, rather derived concept: passing to homology is something like passing to the free symmetric monoidal completion of the archetypical $\infty$-costack living in homotopy theory: the fundamental $\infty$-groupoid. This free symmetric monoidal completio is something like the $\infty$-groupoid where all $k$-cells of the fundamental $\infty$-groupoid may freely and invertibly be formally added. At the same time, homology proper forgets all the original composition operation in the fundamental $\infty$-groupoid.

So from this perspective it appears as a comparatively ad hoc, comparatively unnatural thing to consider. And this may help to reduce the surprise that its dual concept is of no big importance.

(Okay, that said, somebody should now mention the Eilenberg-Steenrod axioms and then I would greatly enjoy a discussion of the relation of cohomology in the sense of these axioms and cohomology in the sense of sheaf-cohomology/nonabelian cohomology etc.)

Posted by: Urs Schreiber on October 22, 2008 10:37 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Urs,

You may be underestimating slightly the role of homology here.

There was an interesting example of an application of homology/cohomotopy some years ago when a problem of von Neumann was solved by Brown-Douglas-Fillmore. They showed that the obstruction to a positive answer was in a Steenrod K-homology group. This was further related to a cohomotopy group. Steenrod homology as such, or Steenrod-Sitnikov homology if you prefer, is a homology for compact spaces that, more or less, forms the system of Cech nerves of open covers, then takes chain complexes, then in addition forms their homotopy limit before taking homology.

This stuff then fed into the K-theory of operator algebras, and so back towards the quantum ideas more usually aired in this café.

I can search out references if anyone is interested.

Posted by: Tim Porter on October 23, 2008 9:45 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

You may be underestimating slightly the role of homology here.

That may be. At least my blatant statement triggered your interesting answer to David’s less blatant questions about the seeming lack of interesting dualities involving cohomotopy! :-)

There was an interesting example of an application of homology/cohomotopy some years ago

[…]

I can search out references if anyone is interested.

I’d be very much interested. Thanks!

Posted by: Urs Schreiber on October 23, 2008 10:33 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

There seems to be a discussion of this in the book on Analytic K-homology by Higson and Roe, but the original work was:

Extensions of $C^*$-algebras, operators with compact self-commutators, and $K$-homology,L. G. Brown, R. G. Douglas, and P. A. Fillmore, Bull. Amer. Math. Soc. Volume 79, Number 5 (1973), 973-978.

There is a substantial literature on this stuff. I think the point is that the cohomology of $C^*$-algebras is related to the Steenrod type homology of the spectrum. But I am no functional analyst. The particular problem of von Neumann was to do with operators that were normal modulo the compact operators and the question was were they necessarily normal + compact. The answer was NO, the obstruction living in one of these homology type groups.

Posted by: Tim Porter on October 23, 2008 6:39 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

…cohomology and homotopy are both fundamental concepts, as follows: cohomology is about gluing and descent of presheaves, hence about $\infty$-stacks, whereas homotopy is about codescent of co-presheaves, hence $\infty$-costacks.

That raises a couple of questions. Is there a notion of co-gluing? And are cohomology and homotopy well named in view of their duality?

Posted by: David Corfield on October 23, 2008 9:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Is there a notion of co-gluing?

Gluing is just another word for descent. So co-gluing is (or should be, I don’t think it is being used much at all so far) just another word for codescent.

I could have titled the entry Codescent and the van Kampen theorem simply Co-gluing.

And the story of the van Kampen theorem exhibts nicely the evolution of the ideas here. It’s quite nice:

let me say in the following “homotopy groupoid” for “fundamental $n$-groupoid” (where $n$ could be some element in $\mathbb{N}$ or could be $\infty$ depending on the precise setup) of a space (since that’s more evocative for our context and maybe would generally be a more suitable term alltogether).

Then: to understand the (vertex $n$-groups of the) homotopy groupoid $\Pi(X)$ of a space $X$ which is covered $\pi : Y \to X$ by a space $Y$, you can compute the homotopy groupoid $\Pi(Y)$ of $Y$ instead (which will be much simpler, for good choices of $Y$) and then co-glue that to obtain the groupoid $Codesc(Y,\Pi)$.

If you do this for $n=0$ in that you take $\Pi = P_0 = Disc(-)$ the operation which sends a space to the disctete category over it, then the co-glued $\Pi$-homotopy groupoid obtained from $Y$ is nothing but the familiar Čech groupoid $Codesc(Y, P_0) = Č(Y) = (Y^{[2]} \stackrel{\to}{\to} Y ) \,.$

This is the weak pushout of $\array{ Y \times_X Y &\to^{\pi_1}& Y \\ \downarrow^{\pi_2} \\ Y }$ and is weakly equivalent to $X$, which itself is the strict pushout. In fact, it is an acyclic fibration over $X$ $Č(Y) \stackrel{\simeq}{\to}\gt X \,.$

So we find that the category valued co-presheaf $P_0 = Disc(-)$ is a co-stack: its value on any space is equivalent to the co-glued values on any cover of that space.

This story continues. When passing from $n=0$ to $n=1$ the above story becomes the story of the van Kampen theorem. Or rather: the van Kampen theorem is concerned with the strict pushout $\array{ \Pi_1(Y \times_X Y) &\to^{\pi_1}& \Pi_1(Y) \\ \downarrow^{\pi_2} \\ \Pi_1(Y) } \,,$ whereas co-gluing of $\Pi_1$s produces the weak pushout $Codesc(Y,\Pi_1)$. Again, we find that the weak co-glued pushout is an acyclic fibration over the strict pushout $Codesc(Y,\Pi_1) \stackrel{\simeq}{\to} \gt \Pi_1(X) \,.$ This is lemma 2.15, p. 15 in Parallel transport and functors.

So: the fundamental 1-groupoid, which is a groupoid-valued co-presheaf $\Pi_1 : Spaces \to Groupoids$ is, too, a co-stack: its value on any space is equivalent to the co-glued values on any cover of that space.

And so on. The fundamental 2-groupoid co-presheaf $\Pi_2 : Spaces \to 2 Groupoids$ is a 2-costack in that its value on any space is equivalent to the co-gluing of its value of any cover of this space. (This is appendix B here).

And are cohomology and homotopy well named in view of their duality?

Good question. Maybe not.

Posted by: Urs Schreiber on October 23, 2008 10:22 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Can I be naughty and suggest that cohomology and homotopy are essentially the same thing?

Classical cohomology is homotopy classes of maps from X to some nice space or spectrum, non-Abelian cohomology is homotopy classes of maps from a space (or (hyper)covering of one) to a n-type. So cohomology is homotopy. In Pursuing Stacks, Grothendieck, if I remember rightly, talked about resolving the domain or coresolving the codomain i.e. the space that is the ‘coefficients’ in cohomology.

There is the slightly bizarre observation that Turaev introduced HQFTs as studying manifolds with extra structure given by a map to a ‘classifying space’. Brightwell and Turner introduced essentially the same construction interpreted as telling us information on the target ‘classifying space’. Both give valid, useful and interesting insights.

One may thus argue that both homotopy and cohomology are a question of comparing two objects, one known with one unknown. The difference is the position, domain or codomain, of the ‘unknown’. Sometimes they coincide!!!!!

Homology is more problematic and seems a lot less motivated, except when we come to homology of algebraic objects, where it can encode very useful behaviour.

Posted by: Tim Porter on October 24, 2008 6:06 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

both homotopy and cohomology are a question of comparing two objects, one known with one unknown.

as I recall, Bott and Tu point this out very clearly early on in their book

Posted by: jim stasheff on October 25, 2008 2:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Can I be naughty and suggest that cohomology and homotopy are essentially the same thing?

Sure. But then allow me to be naughty, too, and insist that they are dual. And that the duality is simly that coming from passing from categories to opposite categories.

Classical cohomology is homotopy classes of maps from $X$ to some nice space or spectrum, non-Abelian cohomology is homotopy classes of maps from a space (or (hyper)covering of one) to a $n$-type. So cohomology is homotopy.

Hm. To be frank, I don’t follow the conclusion here.

First I feel that “homotopy classes of maps” and “homotopy groups” is, while related, two different things. When I think of “cohomology versus homotopy” I am thinking of homotopy groups and their non-abelian generalizations.

And not all cohomology is invariant under homotopy! Of course everything following Eilenberg-Steenrod is (by definition), but there are perfectly respectable things called cohomology which are not. Sheaf cohomology in general is not! For instance Deligne cohomology is not. Even the differential versions of Eilenberg-Steenrod cohomology are not. In fact, whenever the cohomology classifies anythiing “with non-flat connection” it is not homotopy invariant.

(Of course there is the point of view that only generalized Eilenberg-Steenrod is “true” cohomology, by definition.)

So that summarizes my point of view currently. But I am emphasizing it mainly so that you can take it apart. If I am wrong, please tell me why.

resolving the domain or coresolving the codomain i.e. the space that is the ‘coefficients’ in cohomology.

Ah, i see. But wouldn’t there still be a difference then?

Cohomology is the colimit over (hyper)covers of maps out of these covers.

Homotopy should be limit over co-covers out of maps into co-covers.

This is dual. Not the same. No?

I may be mixed up. Please set me straight.

One may thus argue that both homotopy and cohomology are a question of comparing two objects, one known with one unknown. The difference is the position, domain or codomain, of the ‘unknown’.

but it’s still a difference, no?

Mayabe we are just using the words “the same” and “dual” differently, since it seems we perfectly agree on the technical aspects of the phenomenon.

Homology is more problematic and seems a lot less motivate

Ah, good. This seems to be the kind of statement I made above which got this discussion started: homology (and then cohomotopy) is somehow less fundamental than the pair cohomology and homotopy.

Posted by: Urs Schreiber on October 25, 2008 5:24 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Of course, you are right if you want to interpret things in terms of homotopy groups. For me the homotopy groups are very poor invariants and the infinity cat models of homotopy types are much better, thus the 2-type of a space is not specified by the first two homotopy groups, but is by the crossed module (which only determines things up to coboundaries in a cohomology groups etc.) and so on.

The universal property of the fundamental group is not mirrored by the higher homotopy groups yet, if Grothendieck was right, algebraic models for n-types do have such a (lax coherent) universal property and that forms part of the basis of a lot of what we are trying to do.

Experience in other subject areas such as algebraic K-theory seems to show that one should consider the homotopy type as the central object of study with the groups and other structures, pairings, Whitehead products, etc, as being useful observations or probes of that homotopy type. Because of all this the complicated structure of sets of homotopy classes of maps between things’ seems to be to be of more importance than homotopy groups or cohomology groups as they are just’ particular instances of the overall situation. (I am not implying that those objects are not useful, nor that they are uninteresting, merely that for me they are less interesting.)

My reaction to your very valid points on Deligne cohomology, non-flat connections, etc., is that I always sort of expect some homotopy-like structure to be available, and it is the lack of geometric input to homotopy’ that makes it inappropriate. Homotopy still enters sometimes via tools such as simplicial sheaves etc. and there the local information may use homotopy but the global does not.

My comment on Grothendieck’s resolving /coresolving quote was not checked up on. (I’m not sure where in the 650 pages of the manuscript it is !!!!!!)

You say: “Cohomology is the colimit over (hyper)covers of maps out of these covers.

Homotopy should be limit over co-covers out of maps into co-covers.”

The situation should probably be homotopy limits’ rather than limits, but again it is a question of what are we to mean by homotopy’, the homotopy type or some groups extracted using a probing’ functor, such as $[S^n, -]$.

Your talk of co-covers’ seems to be related to Shape Theoretic ideas, where a space was studied using maps into a nice’ space, in that case usually the Hilbert cube. (I can provide more info to anyone who is interested!!)

My problem with classical homology’ is that the Moore spaces are a lot less related to group presentation type ideas than are Eilenberg-MacLane spaces so I end up a feeling of dissatisfaction. Cohomotopy often seems rather silly as there is no real reason for thinking that mapping into a sphere is a good thing to do. (Again that is a personal view, nothing more.) There is the historical use of these terms and they are not optimal from the perspective of this discussion.

There is one thought and that is that often in algebraic cohomology the coefficients are the same as the object being studied and you end up with $H_*(k,k)$ or similar, as being the key object of study. Then is this to be thought of as homotopy’ like or cohomology like. My thought (possibly half baked) is that it is both and neither, but as I have never really studied these I should probably abstain from saying more!

My conclusion is that $[A,B]$ is probably more important to study in general than either $H^*(A,G)$ or $\pi_*(B)$, and that really it is the homotopy types themselves, here of $A$ and $B$, that need studying as directly as possible.

Posted by: Tim Porter on October 25, 2008 6:31 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

For me the homotopy groups are very poor invariants and the infinity cat models of homotopy types are much better, thus the 2-type of a space is not specified by the first two homotopy groups, but is by the crossed module (which only determines things up to coboundaries in a cohomology groups etc.) and so on.

So instead of the copresheaf $X \mapsto \Pi_\omega(X)$, the strict $\infty$-path groupoid, we need to eventually consider the weak fundamental $\infty$-groupoid. I suppose.

Posted by: Urs Schreiber on October 25, 2008 10:37 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I think my point is that you divide out by the minimum for the calculations at that point (and that is partially a matter of taste!) The more baggage you carry around the harder it is to travel but the more likely it is that you have the right equipment when you need it! The problem is trying to optimise the balance point: what to carry v. what to dump. Sometimes I have noticed that initially you go through with an idea at a low level, only to realise that you can lift arguments to the next level up and possibly to all levels at once. That is so imprecise as to be useless perhaps, but dropping down too early to homotopy groups or even to infinity groupoids may destroy the geometric picture.

As an instance of this, you have been mentioning Ronnie and Phil Higgin’s crossed complex theory, which is a beautiful rich theory with a lot of lovely results which can probably be pushed further into the smooth case. There is however a relatively undeveloped theory of 2-crossed complexes which have slightly less Abelian information in them. They are related to Baues’s quadratic complexes. They have, instead of a crossed module at the base, a 2-crossed module, and so can encode a smidgen of the Whitehead product structure of a homotopy type. There is a corresponding infinity category model of course, with a weak interchange at the relevant level. Often it will be sufficient to work with crossed complexes, as the work involved is less arduous than for the next stage up. For general theory it would also be possible to use infinitely structured models with the possibility of Whitehead products etc in all dimensions. Possibly however, to get more calculations and more tractible theory, you may need to drop back to a finite level and it is the interplay of adjacent levels that also gives a lot of useful structural information.

Grothendieck mentioned early on in Pursuing Stacks that there would be as many models for n-types as there are mathematicians working with them as the exact choice would be a question of the individuals intuitions etc. I got the impression that he also meant that it would depend on the task the mathematician wanted that model to do. He accepted for instance that simplicial models for things were powerful but he did not like them probably because they were very large. What model suits in the smooth context is thus very difficult to gauge.

Posted by: Tim Porter on October 26, 2008 8:55 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

2-crossed complexes should be to groupoids in $(\omega Cat, \otimes_{GrayCrans})-Cat$ as crossed complexes are to groupoids in $\omega$-categories. Hence they know about the “first level” of non-strict composition.

Posted by: Urs Schreiber on October 26, 2008 11:05 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

That feels’ right, but I would need to check details.

At the risk of repeating myself, I do think that the links between the $\infty$-Cat panoply and the machinery of Whitehead products etc. is an area that could, and perhaps should, be looked at in detail in the near future. The Peiffer lifting’ map, (Conduché), does correspond to the Gray-tensor structure by old work of Joyal and Tierney, if I remember rightly, and it also links with the corresponding Whitehead product formula, but there I forget the exact formula. Morally’ they are more or less the same in the relevant dimension (i.e. covering the interchange law.)

At the stack/costack level then the various homotopy operations ssuch as Whitehead products should be realisable by interesting pairing operations.

Posted by: Tim Porter on October 27, 2008 9:53 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

where do you see anything like Whitehead products in that infty-cat panolply?

Posted by: jim stasheff on October 27, 2008 2:15 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

The oldest weak infinity category model could be considered to be Kan complexes and thus simplicially enriched groupoids and there there are clear formulae for the Whitehead products as an ordered product of shuffle based terms given in an unpublished result of Kan. I do not know of a comparable result for other, say globular, models. (I have written out a proof of Kan’s formula if anyone is interested. The formula itself is on page 197 of the 1971 article of Curtis.)

Limiting the scope, in low dimsnional 2-type models (say Conduché’s 2-crossed modules), as I may have mentioned earlier, the Whitehead product can be linked to the Peiffer lifting and thus essentially to the lack of interchange in the corresponding categorical model. I seem to remember that Marco Grandis had some results in this direction, but am not sure. I know Ronnie Brown has a paper in which a formula linking the Peiffer lifting / h-map of a crossed square with the Whitehead product. (Again I can give a reference if I look hard enough or if I ask Ronnie!)

Slightly away from n-categorical models as such Baues introduced quadratic complexes and in his book on 4-dimensional complexes gave a formula involving Whitehead products, their definition being in his book on Algebraic Homotopy. Those quadratic complexes are modified versions of Conduché’s ones. The fact that his approach uses suspensions makes me wonder it John and Jim Dolan ever considered Whitehead products in their work.

My guess is that the Whitehead product formulae should be measure of lack of interchange at the various levels.

There was a talk given at one of the Como meetings by Florence Marty which looked at 3-categories following some ideas of Olivier Leroy. Her thesis title (1999) had been:

Approche en dimension supérieure des 3-catégories augmentées d’Olivier Leroy

I felt that some of the ideas she had put forward were possibly going in the direction of higher pairings related to Whitehead products, but I may be wrong… and perhaps someone else may know of this as well and can provide me with an an answer. There were various p + q interchanges evident in that stuff, and I am sure that in the other models for weak infinity cats there should be, either evident or lurking, similar structures.

Posted by: Tim Porter on October 27, 2008 3:46 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Tim wrote: My guess is that the Whitehead product formulae should be measure of lack of interchange at the various levels.

Samelson products seem more reasonable in
monoidal cats. Topologically Whitehead and Samelson are equivalent but in a cat context
would Whitehead obtsin at the nerve level?

Posted by: jim stasheff on October 27, 2008 6:06 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Samelson products are more appropriate unless one is working with n-groupoid type objects. I was being sloppy in my terminology, sorry!

Does anyone know of other references for Samelson type products, say in monoidal categories as Jim suggests.

Posted by: Tim Porter on October 27, 2008 9:23 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

The paper of Ronnie’s is

74. “Computing homotopy types using crossed $n$-cubes of groups”, {\em Adams Memorial Symposium on Algebraic Topology}, Vol 1, edited N. Ray and G Walker, Cambridge University Press, 1992, 187-210.

The number refers to the numbering in his publication list at

http://www.bangor.ac.uk/~mas010/publicfull.htm

which contains a link to a pdf file of the paper. The result to note is his theorem 2.4 on page 199. The proof is given later in the paper.

Posted by: Tim Porter on October 27, 2008 6:07 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

or coresolving the codomain i.e. the space that is the ‘coefficients’ in cohomology.

At the risk of blowing my own horn, I once figured out how to define cohomology with coefficients in a simplicial topological group, by doing just this - leaving the space in question alone, but forming a resolution along the lines of Segal in “Cohomology of topological groups” 1970 Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69). I got some nice monad, but then lack of experience rose up and stopped me from doing anything interesting with it.

Posted by: David Roberts on October 28, 2008 1:54 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

resolution of ____ as ____?

Posted by: jim stasheff on October 28, 2008 3:58 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

There is a monad on simplicial topological groups such that the unit of the monad makes a simplicial topological group a subgroup of a contractible one. The cosimplicial resolution of the simplicial topological group is made up of contractible simplicial topological groups.

Maybe I was fooling myself and got something uninteresting, but since $\infty$-things are popular here now, I thought it was worth dusting off. Certainly a smooth version would be interesting, as well as the links to $L_\infty$-algebras.

Posted by: David Roberts on October 28, 2008 6:42 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Interesting to see it done monadicly but
does this differ from the WG simplicial constuction giving the universal
G –> WG –> \bar WG
?

Posted by: jim stasheff on October 28, 2008 3:21 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Here is what I do know (as against the mass of things in this area that I think might be true, or that I don’t know that I don’t know (shades of Rumsfeldisms!))

Denote the adjoint of TotDec by $\nabla$. Take a simplicial group, $G$, apply nerve to all dimensions, you get a bisimplicial set. So naturally you immediately feel the urge to apply $\nabla$. Giving into your baser instincts you do not resist and so calculate $\nabla (Ner(G))$ and to your amazement you find it is our old friend the classifying space $\bar W G$.

Now I cannot remember how to get $WG$ itself as such a neat object except in the case where $G$ is simply a group when it can be given as a homotopy colimit. Does anyone have a good way in general to view this?

Posted by: Tim Porter on October 28, 2008 5:03 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

That is where I come in. The underlying simplicial set of the contractible simplicial group $\mathcal{E}G$ into which the simplicial group $G$ embeds is isomorphic to $WG$. At each level they are made up of the same sets, but the face and degeneracy operators, and the map to $\bar{W}G$, are different. $WG$ and $\mathcal{E}G$ are also isomorphic as $G$-bundles, it’s just that $\mathcal{E}G$ acts like the universal bundle Urs and I constructed for 2-groups.

Consider now the simplicial group $G$ which is just an ordinary group. The difference I find is nicely illustrated by comparing the (nerve of the) strict 2-group given by the crossed module $id:G \to G$ (called $Inn(G)$ in various places by Urs and me), and the (nerve of the) boring old pair groupoid $disc(G)$ on $G$. The latter can be given a group structure by making the arrows the product group $G \times G$. The higher simplices of $NInn(G)$ are the iterated semidirect products of $G$, whereas the higher simplices of $Ndisc(G)$ are just the product groups. Both of these nerves are isomorphic simplicial groups, but it is the second one which can be generalised to general simplicial groups with a lot less effort (consider the paper “The inner automorphism 3-group of a strict 2-group”, which repeats the previous example for simplicial groups which are the nerves of strict 2-groups - that was hard enough!)

One thing that doesn’t seem to work is that one can form $\mathbb{E}M$ for a simplicial monoid, but this doesn’t have a map to $\bar{W}M$ (this object makes sense, the inverse not being utilised in the definition).

Posted by: David Roberts on October 29, 2008 1:14 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Also, if one applies first the functor $\mathbf{sTopGrp} \to \mathbf{sTopGrp}$ underlying the monad, and then the Moore complex functor (which takes simplicial groups to (nonabelian) complexes of groups), the resulting thing looks like the mapping cone of the identity map. This as you know links in very much with the stuff I did with Urs for strict 2-groups.

Posted by: David Roberts on October 28, 2008 6:48 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

A useful phrase in this sort of blog is That reminds me of…’ David’s contributions remind me of’ some results that I learnt from my ex-student Phil Ehlers, who in turn learnt them from Jack Duskin and Don van Osdol. They are mentioned in Duskin’s Memoirs AMS volume. I have been told they go further back to comments by Bill Lawvere, but do not have the reference. They form part of a categorically well known situation, but that is not quite relevant here so … .

The decalage functor on simplicial sets strips off the last face and last degeneracy, (or the zeroth ones depending on your conventions). This gives a split augmented simplicial set. There is an adjoint that throws away the splitting and the augmentation. This adjoint pair forms a category of algebras on SAS (= split augmented simplicial sets) and that (Eilenberg Moore) category of algebras is equivalent to the category of simplicial sets. (You can replace sets here by anything).

The corresponding comonadic resolution of a simplicial object $K$ gives a bisimplicial object, which is $TotalDec(K)$, and as mentioned yesterday is the composition of the functor corresponding to $K$ with the ordinal sum from the pairs of ordinals to $\Delta$.

The total decalage and other higher order versions were one of the tools that I used in my algebraic proof of Loday’s theorem on algebraic models for $n+1$-types. (I can provide more details and references if it would help) The idea of using that machinery for simplicial topological groups sounds a great one.

Posted by: Tim Porter on October 28, 2008 8:15 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

The corresponding comonadic resolution of a simplicial object K gives a bisimplicial object, which is TotalDec($K$)

here is where my version is different: I get a cosimplicial simplicial object (is there a short name for this??). Duskin’s use of Dec is to form a resolution of the space, not the coefficients.

Also, the inclusion to a contractible thing is in my case the unit of the monad, whereas with Dec it is a splitting of the counit of the comonad.

On the link to twisted cartesian products: this is fine when dealing with simplicial groups in Set, but in the topological setting this theory is wholly inappropriate (mostly due to the fact that forgetting the group structure doesn’t supply one with a plentitude of sections of epimorphisms).

Posted by: David Roberts on October 29, 2008 1:16 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

David R. wrote, on the resolution of simplicial topological groups he mentioned #:

Also, if one applies first the functor $\mathbf{sTopGrp} \to \mathbf{sTopGrp}$ underlying the monad, and then the Moore complex functor (which takes simplicial groups to (nonabelian) complexes of groups), the resulting thing looks like the mapping cone of the identity map. This as you know links in very much with the stuff I did with Urs for strict 2-groups.

Interesting. So does this resolution for the case of a plain ordinary group amount to the injection $\mathbf{B}G \hookrightarrow \mathbf{B}\mathbf{E}G$ ?

Let me see. It seems you said that maps from spaces into the resolution gave you $G$-cohomology. That seems to suggest the resolution is more like $G Tor$, the category of $G$-torsors and $G$-equivariant maps, with the canonical inclusion $\mathbf{B}G \hookrightarrow G Tor$.

Ah, that makes we think the following:

remember that square diagram I am so fond of, which gives local trivialization of functors $\array{ \mathbf{Y} &\to \gt& \mathbf{X} \\ \downarrow &\Downarrow^\simeq& \downarrow^{tra} \\ \mathbf{B}G &\hookrightarrow& G Tor } \,.$

It can be seen as one way of talking about anafunctors from $\mathbf{X}$ to $\mathbf{B}G$.

Now, the top horizontal arrow is a resolution of the domain $\mathbf{X}$. Now maybe what you are saying suggests that I think of the lower horizontal arrow as a co-resolution of the codomain $\mathbf{B}G$.

That would mean that the transport functor $tra : \mathbf{X} \to G Tor$ would have to be regarded as a morphism from the domain into a coresolution.

Hm. I need to think about this. I always felt that there must be more to the fact that in this diagram we have a fibration on top and an injection at the bottom, where both of them can be taken as weak equivalences, hence as replacements.

Posted by: Urs Schreiber on October 28, 2008 8:48 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I’m notationally challenged
BG subset of EG??

versus the old fashioned? G –> EG –> BG

and G-Tor versus equiv classes of G bundles?/

Posted by: jim stasheff on October 28, 2008 3:26 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

BG subset of EG?

Oops, that was a typo (have fixed it now). It should have read $\mathbf{B}G \hookrightarrow \mathbf{B}\mathbf{E}G \,,$ which is just the familiar inclusion $G \hookrightarrow \mathbf{E}G$ but noticing that there is really a group structure on $\mathbf{E}G$ and that with respect to this this a an ($n$-)group homomorphism.

and G-Tor versus equiv classes of G bundles?

Yes, not equivalence classes. As a category in Sets $G Tor$ is supposed to have as objects principal $G$-sets and as morphisms equivariant maps. So a smooth version of $G Tor$ assigns to a test domain $U$ the category of principal $G$-bundles over $U$.

Posted by: Urs Schreiber on October 28, 2008 4:10 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

On the idea that $\mathbf{B}G \to G-Tor$ is a coresolution: there is stackification involved. More precisely, the category of all $G$-bundles is a stack on $\mathbf{Top}$, and this is the stackification of the constant presheaf $\mathbf{B}G$. $G-Tor$ is then the fibre of this stack over the terminal object (i.e. $pt$).

Posted by: David Roberts on October 29, 2008 1:16 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

On the idea that $\mathbf{B}G \to G Tor$ is a coresolution: there is stackification involved.

Yes, that’s exactly how I described it in a previous comment, which for some reason is now sitting below the present lines, namely here.

But do you know if the idea of addressing maps into the codomain of the inclusion of the prestack into its stackification as mapping into a coresolution is appropriate?

Actually, it could well be, because with respect to the model structure on $Sheaves(..,\omega Cat)$ the inclusion $\mathbf{B}G \to G Tor$ is a local weak equivalence: for every test domain and every point in the test domain there is a neighbourhood of that point such that restricting to the morphism of $\omega$-categories over that open neighbourhood produces a weak equivalence of $\omega$-categories – which is just another way of saying that $G$-bundles are locally trivial.

Posted by: Urs Schreiber on October 29, 2008 7:17 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Inspired by David’s comment I wrote

That would mean that the transport functor $tra : \mathbf{X} \to G Tor$ would have to be regarded as a morphism from the domain into a coresolution.

Indeed, there is a nice general nonsense kind of way to fill this with life:

for $G$ an $\omega$-group internal to Spaces = Sheaves(CartesianSpaces), the internal way to say $G Tor$ is to take $G Tor$ tautologously to be the $\omega$-category valued sheaf which to test domain $U$ assigns $G Tor : U \mapsto H(U,\mathbf{B}G) \,,$ where $H(U,\mathbf{B}G)$ is the cohomology $\omega$-category we have been talking about, i.e $H(U,\mathbf{B}G) = colim_{Y \to U} Desc(Y, hom(-,\mathbf{B}G))$, the “$\omega$-stackification” of trivial $G$-principal bundles.

Anyway, we have a canonical inclusion $\mathbf{B}G \hookrightarrow G Tor$ this way, coming on each test domain by the inclusion of trivial $G$-bundles into all $G$-bundles using Yoneda once. And this should be a local/stalkwise cofibration. And sure enough we get $Hom(\mathbf{X}, G Tor) \simeq H(X,\mathbf{B}G)$ again just by Yoneda. This we can read as saying that, indeed, $G$-cohomology on $\mathbf{X}$ is obtained by maps from $\mathbf{X}$ into coresolutions of $\mathbf{B}G$.

Posted by: Urs Schreiber on October 28, 2008 9:18 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

With reference to this point, it may be worth noting that Jack Duskin in his AMS memoir seems to imply that the Dec-based triple’ cohomology was giving the twisted cartesian product / simplicial fibre bundle theory when one looked at the cohomology of a simplicial set with coefficients in a simplicial group. I never fuller checked this out.

Posted by: Tim Porter on October 28, 2008 11:47 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

[…] Jack Duskin in his AMS memoir […]

By the way, the thesis by Igor Bakovič, which I once announced here is now available:

Igor Bakovič: Bigroupoid 2-torsors.

He discusses 2-bundles with general bigroupoids as “structure groups”.

In the last section of his thesis he shows that taking their nerves these reproduce Duskin’s simplicial definition in degree $n=2$.

Interestingly, he does so using the notion of action bigroupoids, generalizing ordinary action groupoids, and shows along the way that with this definition David R.’s and mine $\mathbf{E}G = INN(G)$ is indeed the action bigroupoid of the 2-group $G$ acting on itself, as one would expect.

I think that if you take all this together, it fits into the picture which we have been drawing, which says that every principal $G$-bundle for $G$ an $\infty$-group with cocycle given by an $\infty$-anafunctor $g : \mathbf{Y} \to \mathbf{B}G$ arises as pullback along this cocycle of the universal $\infty$-bundle $\mathbf{E}G \to \mathbf{B}G \,.$

Posted by: Urs Schreiber on October 28, 2008 12:16 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Urs wrote: which says that every principal G-bundle for G an ∞-group with cocycle given by an ∞-anafunctor g:Y→BG arises as pullback along this cocycle of the universal ∞-bundle
EG→BG.

what’s wrong with the classical version uped to the infty-level:
every principal G-bundle for G an ∞-group
over a space Y arises as pullback along a map Y –> BG of the universal ∞-bundle
EG→BG.

Posted by: jim stasheff on October 28, 2008 3:31 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Urs wrote:

which says that every principal $G$-bundle for $G$ an $\infty$-group with cocycle given by an $\infty$-anafunctor $g : \mathbf{Y} \to \mathbf{B}G$ arises as pullback along this cocycle of the universal $\infty$-bundle $\mathbf{E}G \to \mathbf{B}G$.

what’s wrong with the classical version uped to the infty-level: every principal $G$-bundle for $G$ an $\infty$-group over a space $Y$ arises as pullback along a map $Y \to B G$ of the universal $\infty$-bundle $E G \to B G$.

I added in again the distinction between boldface and not, because that’s crucial here:

I write $\mathbf{B}G$ for the one-object $\infty$-groupoid corresponding to the $\infty$-group $G$.

I write $\mathbf{E}G$ for the $\infty$-groupoid arising as the pullback $\array{ \mathbf{E}G &\to& (\mathbf{B}G)^I \\ \downarrow && \downarrow^{dom} \\ pt &\to& \mathbf{B}G } \,.$

A $G$-principal $\infty$-bundle over a space $X$ is an $\infty$-groupoid $P$ with a map down to $X$, $P \to X$ (with $X$ regarded as a discrete $\infty$-groupoid) such that for some cover $\pi : Y \to X$ there is a $G$-equivariant equivalence $t : \pi^* P \stackrel{\simeq}{\to} Y \times G$ of $\infty$-groupoids.

Interpreting all this in some suitable context, it should be true that any such choice $t$ gives rise to an $\infty$-functor $g$

$\array{ Codesc(Y) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\simeq} \\ X }$ such that the original $\infty$-bundle $P \to X$ is equivalent to the pullback $g^* \mathbf{E}G \to X$ in $\array{ g^* \mathbf{E}G &\to& \mathbf{E}G \\ \downarrow && \downarrow^{codom} \\ Codesc(Y) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\simeq} \\ X } \,.$

You are asking for a “classical” picture:

one can take geometric realizations of everything in sight here and send all this from the world of $\infty$-groupoids to the world of topological spaces. Then the $\infty$-group $G$ becomes a topological group $|G|$ and we should have $|\mathbf{B}G| \simeq B |G| \,.$ But showing that this equation is true requires work.

For topological 2-groups satisfying some condition it has been shown, see here.

Posted by: Urs Schreiber on October 28, 2008 4:30 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Warning:

the link to Igor’s thesis which I provided # seems to currently point to an old version. The relation to $INN(G)$ etc. which I mentioned is in the official end version which I have as a pdf file, but does not seem to be in the version which is currently on the web.

I have contacted Igor. I’ll let you know when the official version is available.

Posted by: Urs Schreiber on October 28, 2008 1:00 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

This blog whizzes by blindingly fast sometimes, but while on the topics of “This reminds me of” and “not meaning to toot my own horn”: the left adjoint to decalage on simplicial objects is like taking the mapping cylinder of the projection $\pi: X \to \pi_0(X)$, i.e., taking the sum of all cones over connected components [as mentioned in my post on the bar construction]. I somehow find “coning” is much more evocative than “decalage”.

In the same vein: the left adjoint to total decalage should be likened (at least in the connected case) to taking the simplicial join of spaces (the space of all line segments between $X$ and $Y$). This may be a useful geometric intuition.

Posted by: Todd Trimble on October 28, 2008 1:20 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Todd,

The evocativeness is dependent on the mother tongue. Décalage is a good term for shifting and then taking the last face and degeneracy maps off as it invokes (in French) amongst other things shifting’ or offset’ (décalage horaire is time difference).

The idea of join in this context is a good and useful one. It has been partially explored, initially by Phil Ehlers in his thesis in 1993 and then I eventually wrote up a version of that part in a short paper:

Joins for (augmented) simplicial sets, Jour. Pure Applied Algebra, 145 (2000) 37-44.

As you suggest it may be a good one to keep hold of for intuition. I think there may be a lot more to say about it, but I’m not sure what!

Posted by: Tim Porter on October 28, 2008 2:02 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

The evocativeness is dependent on the mother tongue…

I know that. I really meant “geometrically evocative”.

Posted by: Todd Trimble on October 28, 2008 2:14 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Then I almost agree, but would add that the dec itself looks like an object of paths starting at base points, i.e. a sort of cocne (like a mapping cocone.)

Linguistic points aside, the left adjoint to total dec gives a functor from bisimplicial objects to simplicial ones related to the codiagonal on bicomplexes. Do you interpret that as a cone? My interpretation of things means that it starts off in the wrong category. Sorry for being thick on this.

The individual dec has an adjoint which is a cone type construction, but I do not see the total dec as doing that.

Posted by: Tim Porter on October 28, 2008 2:46 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Tim wrote:

the left adjoint to total dec gives a functor from bisimplicial objects to simplicial ones related to the codiagonal on bicomplexes. Do you interpret that as a cone?

I interpret that as a simplicial join. Oh, okay, I see what you mean, and you’re right; here’s what I really meant. Thinking of

$Set^{\Delta^{op} \times \Delta^{op}}$

as a free cocompletion of $\Delta \times \Delta$, there’s a universal separately cocontinuous functor

$Set^{\Delta^{op}} \times Set^{\Delta^{op}} \to Set^{\Delta^{op} \times \Delta^{op}}$

and when you compose that with the left adjoint to total decalage, you get a version of simplicial join (I say a version because to make it literally true, you have to restrict to connected objects – it’s really the sum of simplicial joins, taken over all pairs of connected components, I think).

Hope that sounds more sensible now.

Posted by: Todd Trimble on October 28, 2008 4:08 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Phil’s result may clarify the situation if I explain it more fully. First I may have got confused with the right and left adjoints of TotDec. The right adjoint is the $\nabla$ which I sometimes call the codiagonal.

Using the Dec on augmented simplicial sets, (ASS) (and not necessarily just on those augmented to the $\pi_0$) define an internal hom $[X,Y]$ by $[X,Y]_m = ASS(X,Dec^{m+1}Y)$. There is a corresponding tensor product given with the obvious isomorphism. There is an coend formula for this$X\otimes Y \cong \int^{p,q}(X_p\times Y_q)\cdot \Delta([p]\oplus [q])$This then interprets as a join, and thus for instance if $X_{-1} = *$, then $X\otimes \Delta[0]$ is the cone on $X$.

That is the same formula as you give I think.

Posted by: Tim Porter on October 28, 2008 5:36 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Tim wrote:

There is an coend formula for this $X \otimes Y \cong \int^{p,q}(X_p \times Y_q)\cdot \Delta([p]\oplus[q])$ This then interprets as a join, and thus for instance if $X_{−1} = *$, then $X \otimes \Delta[0]$ is the cone on $X$. That is the same formula as you give I think.

Yes, that’s exactly what I had in mind. Good to know it’s in the literature; thanks for the citation.

With all due respect to Tom Leinster, I generally find it more convenient to work with augmented simplicial objects than simplicial ones, and I think Phil is quite right to cast it in that context.

Posted by: Todd Trimble on October 28, 2008 6:10 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

There is a coend formula for this $X \otimes Y \simeq \int^{p,q} (X_p, Y_q) \cdot \Delta([p] \oplus [q])$

Is there a similar coend formula if not for the Crans-Gray tensor product itself then maybe for a cofibrant replacement?

For $C$ an $\omega$-category, hitting it first with $\omega$-nerve and then with the left adjoint yields $\int^n Hom(O(\Delta^n),C)\cdot O(\Delta^n)$ and I originally thought this ought to be essentially the cofibrant replacement of $C$ (though I was warned that this idea looks wrong).

Now for $C$ and $D$ two $\omega$-categories one could consider

$\int^{p,q} Hom(O(\Delta^p),C)\times Hom(O(\Delta^q),D) \cdot O(\Delta^{(p+q)}) \,.$

Is that at all related to $C \otimes_{GrayCrans} D$ ?

Posted by: Urs Schreiber on October 28, 2008 8:26 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Dear Urs,

There is a question that’s been bothering me for quite some time and perhaps this thread is a reasonable place to ask it. Is there a generalization of Poincare duality to non-abelian cohomology? That is, take a locally constant sheaf U of non-abelian groups on some reasonable manifold X. Is there a cohomological description of a ‘dual’ for $H^1(X,U)$? I put the dual in quotes for the obvious reason.

Posted by: Minhyong Kim on October 25, 2008 12:48 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Is there a generalization of Poincaré duality to non-abelian cohomology?

This is a great question. Alas, I don’t know. I was remined of the importance of this just recently by Chris, André and Michael #.

But I don’t know. I’ll try to keep it in my mind though. When I run into anything potentially useful, I’ll drop a note.

Posted by: Urs Schreiber on October 25, 2008 5:05 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

The problem of duality in non-Abelian cohomology is difficult. Sheaf cohomology of any sort is based on Cech type methods and Cech cohomology as such is dual not to Cech homology but to Steenrod-Sitnikov homology, but this is not sheaf based.

Non-Abelian homology of groups based on the non-Abelian tensor product of Brown and Loday has been explored by various people, but it is not clear that it provides any insights that help at the sheaf theoretic level.

Ronnie Brown and I over a period of years tried to look at Huebschmann’s crossed extension based representations of cohomology classes in the hope of somehow finding a dual, but we did not see how to do it.

Does anyone have some new insights that we did not have? I am also very interested in this.

Posted by: Tim Porter on October 25, 2008 5:20 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

One possible answer is that nonabelian Poincare duality is the geometric Langlands correspondence.

Let me try to justify this very roughly. (I might be saying total nonsense, but why not? usually one thinks of geometric Langlands is a kind of nonabelian Pontrjagin/Fourier duality, but aren’t all dualities the same??)

First if we want a fully nonabelian duality I would assume we have to be in two dimensions, since we want H^1 to be dual to H^1 (the higher cohomology groups being effectively abelian - we could take coefficients to be more complicated homotopy types than groups, but still the only truly nonabelian part appears in H^1, or H^0).

So we are studying self-duality of H^1 on a Riemann surface. Now in the number field case, Artin and Tate I think (I’m sure you know this much better) explained that class field theory can be seen as a form of Poincare duality. In the geometric setting, geometric class field theory comes down to the selfduality of the Jacobian. To see this a little more clearly, let T be a torus and T^ the dual torus. Then Poincare duality for H^1 with
Lie(T) coefficients has an exponentiated analog, which is that (the identity components of) H^1(X,T) and H^1(X,T^) are dual abelian varieties (the Jacobian is the case of the one-dimensional torus).

Geometric Langlands is then a kind of Poincare duality for nonabelian H^1 in the case that the coefficients are a reductive group. Very roughly this says that H^1(X,G) and H^1(X,G^) (where G^ is the Langlands dual group) are “dual”. Here duality has many meanings:

1. their cotangent bundles form dual integrable systems. (this is a kind of duality for nonabelian Dolbeault H^1)

2. their categories of twisted D-modules
are equivalent, for reciprocal twistings.
If we think of these categories as categorified analogs of de Rham cohomology, this looks very much like a kind of Poincare duality.

I’m happy to try to elaborate if this is useful (and if I don’t realize in 5 minutes I’ve gotten all my duals wrong).

Posted by: David Ben-Zvi on October 26, 2008 12:02 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Thanks very much to all three of you for the replies. I’ll go out on a limb and formulate a rather grandiose prediction: a good answer to this question of duality should lead to fundamentally new breakthroughs in number theory. If n-categorical matters contribute to this, I would be really delighted.

I had looked earlier into the work of Ronnie Brown and Tim Porter without absorbing more than a superficial overview, so I’m glad to hear that this question occurred to them. I presume it doesn’t help at all to restrict to spaces where everything can be interpreted in term of group cohomology?

David, what you suggest has been very much on my mind, but I can’t seem to do much with it. But regardless of my own motivation, it seems worthwhile to pursue this line of thought even just for a local field, which, you’ll recall, is like the deleted tubular neighborhood of a circle inside a three-manifold, and hence, like a two-manifold. The vague but reasonable question is: Is there a cohomological formulation of local Langlands’ duality? As you point out, local Tate duality for Galois cohomology is equivalent to local class field theory, so it seems natural enough to ask for a reformulation of non-abelian class field theory in cohomological form in a way that is manifestly a duality. Recall that the usual sense of the term ‘Langlands’ duality’ over local fields or number fields is mostly very philosophical. As you might have guessed, it’s actually this kind of setting that motivated my inquiry about Poincare duality.

Here are a few comments/questions about your proposal in the case of a Riemann surface $X$ over $C$. First of all, let me clarify notation by stating clearly that I denote by $H^1(X,G)$ the *locally constant* cohomology, so that if $G$ is $GL_n$, then this is the cotangent bundle to the moduli space of vector bundles. As you say, $H^1(X,G)$ and $H^1(X,G^d )$ form dual Hitchin fibrations.

Here then are three questions:

1. Is this formalism really well-developed for locally constant $G$ as opposed to *constant* $G$? Where can I find this?

2. Is there a construction of this sort for unipotent groups $U$? Note that the usual functions that define the Hitchin fibrations will all be constant. Of course, this question subsumes the one that asks what the Langlands dual of $U$ is. If $U$ is just a vector group, presumably it’s just the usual dual. But what if it’s, for example, a Heisenberg group? Using the symplectic pairing on the vector group quotient, it rather feels like the Heisenberg group itself should be self-dual, in some sense. Of course, I would also like to move up to unipotency of higher order.

Incidentally, the question of full duality subsumes a weaker statement that is addressed, in a vague sense, by the theory of Hitchin fibrations for reductive groups: How can one naturally construct’ functions on $H^1(X,G)$? For unipotent groups, it’s hard to imagine any functions at all.

3. Is there a description of the Hitchin fibration that doesn’t use differential forms? Do you know a way of working directly with group co-chains or some categorical refinements?

If you have any insight on these questions, even very vague ones, I’d be very grateful to hear of them.

Posted by: Minhyong Kim on October 27, 2008 3:36 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

A thought occurs to me about this question of non-Abelian cohomology and possible duality. It is not fully baked’ perhaps not even half-baked but here it is anyhow.

In many (even most) approaches to non-Abelian cohomology, the `real’ object that is looked at is some sort of category of non-Abelian extensions or similar. This category then yields various invariants such as its set of components. (Work in a suitable topos if you like and look at Breen’s various papers as well as Jack Duskin’s) If one was to have a duality theory, what would this mean at this category level? If categorification is important in this then we should take the category to be more fundamental than the cohomology set, pointed set, group or whatever and then it surely must be possible to give some interpretation (in cases where Poincaré duality is known to work) of that duality at the category of extensions level. Does that then give us something when no duality is yet known?

Posted by: Tim Porter on October 27, 2008 6:45 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I’ll try to post a better reply later, but I thought I would express full sympathy with this view. What is a good interpretation of duality at the categorical level? Perhaps one could ask for an interpretation of cup-products in the same vein. Incidentally, when discussing $H^1$, I find it somewhat important to think of them as torsors rather than extensions.

Posted by: Minhyong Kim on October 29, 2008 11:44 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Minhyong, These are very interesting questions you pose. Let me note one thing first: the locally constant cohomology $H^1(X,G)$ (or nonabelian Betti $H^1$) as an algebraic variety (or stack) doesn’t admit a Hitchin fibration - the Hitchin map is only algebraic in a different algebraic structure, namely as nonabelian Dolbeault $H^1$ (moduli space of Higgs bundles). That’s one reason it’s hard to see how to write algebraic functions. If we’re taking the Betti structure as you suggest, rather than the deRham or crystalline $H^1$ (which is the same as a complex manifold but not algebraically), then we do have a lot of algebraic functions given by applying invariant polynomials of $G$ to the monodromies along loops in the Riemann surface (I think these often go by the name Goldman functions). But these are of quite a different nature. For nonreductive groups I don’t think you have a version of the nonabelian Hodge theorem, so it’s hard to know how to compare the Hitchin type functionals you have on the Dolbeault space with something on the Betti space. But maybe you just want to consider the Dolbeault space? but if there aren’t many invariant polynomials on the Lie algebra, as you point out, there aren’t interesting Hitchin functionals.

One general problem with what I suggested (which might just be a problem with a nonabelian Poincare duality, I don’t know) is that the Langlands duality is far from a formal operation. In particular I don’t think it has an analog for nonreductive groups. For example I think the Hecke theory for unipotent groups will be trivial, so you get a trivial answer as to what the dual group should be. Of course another kind of (Fourier) dual for a unipotent group is suggested by the orbit method – it is the dual to the Lie algebra of the group, but considered not just as an abelian group, but rather with the coadjoint action - this is the closest substitute I know for the Langlands dual in this case. Maybe if one understood this far more deeply than I do one could use it to formulate the kind of duality you want.

I’ve tried to formulate even what you would assign to a finite group, but gauge theory seems to suggest that it be self-dual, rather than seeing an interesting Cartier-Pontrjagin-Fourier type duality. (The argument was: look at Hecke operators on moduli of G bundles for G finite - these are given by gauge theoretic reasoning by sheaves on the moduli of G bundles on the two-sphere, but these are just representations of G, not of some cool new dual!)

So that’s a long way to say: I have no idea! my read would be that Langlands duality is a miracle and we should be thankful for having even that, but of course another read is that I’ve totally missed the point of this Poincare duality.

Posted by: David Ben-Zvi on October 28, 2008 4:00 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

….cohomology and homotopy are both fundamental concepts, as follows: cohomology is about gluing and descent of presheaves, hence about oo-stacks, whereas homotopy is about codescent of co-presheaves, hence oo-costacks.

Through a mirror darkly? which is it that is fundamental? cohomology and homotopy or stacks and costacks?

Posted by: jim stasheff on October 23, 2008 3:20 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

which is it that is fundamental? cohomology and homotopy or stacks and costacks?

$\infty$-Stacks and $\infty$-costacks are fundamental.

And $\infty$-stacks are all about cohomology (in the sense of sheaf cohomology, which is a bit different from that of generalized Eilenberg-Steenrod cohomology):

an $\infty$-stack with values in abelian $\infty$-groups is essentially a sheaf of complexes of $\infty$-groups which is equivalent to the cohomology with values in itself.

Dually, $\infty$-costacks are about homotopy.

Posted by: Urs Schreiber on October 23, 2008 7:56 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

There’s more to the story than that, though that is relevant. The language I would prefer is to refer to the n-connected cover X of a space X which has the property
that \pi_k(X) = 0 for k \leq n AND
there is a map X –> X inducing isomorphisms \pi_k(X) to \pi_k(X) for k > n. How to construct such spaces is of interest but often irrelevaant to using them.

Posted by: jim stasheff on September 25, 2008 1:59 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I might need help with the following:

what is the relevance of the continuity of the section $a : G \to P G$ in a neighbourhood of the neutral element?

It certainly resonates with what I have seen before. In the construction used by Brylinski-McLaughlin # one considers even smooth such sections, patchwise. There I know what it is good for, since there the resulting simplices in $G$ are further sent to elements in $U(1)$ by integrating smooth forms over them.

Posted by: Urs Schreiber on September 23, 2008 10:30 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I’ll say some random stuff that might be slightly useful.

In group cohomology as traditionally formulated, a cochain is a map $f : G^n \to A$. Here $G$ is a group, and $A$ an abelian group on which $G$ acts. There’s no requirement that $f$ be continuous or smooth. Indeed, there’s no way to formulate such a requirement.

When $G$ and $A$ are topological groups and the action of $G$ on $A$ is continuous, we can require that our cochains $f$ be continuous. This gives a new kind of group cohomology with strikingly different properties. It’s usually called ‘van Est’ cohomology, since it was first studied here:

• W. T. van Est, On the algebraic cohomology concepts in Lie groups I & II Indag. Math. 17 (1955), 225–233, 286–294.

Similarly, when $G$ and $A$ are Lie groups (or more general smooth groups) and the action of $G$ on $A$ is smooth, there’s a kind of group cohomology where the cochains are required to be smooth.

This is often also called van Est cohomology, since van Est showed that for $G$ and $A$ Lie groups, the group cohomology defined with continuous cocycles matches the group cohomology defined with smooth cocycles. Both are very different from group cohomology as defined using arbitrary cocycles!

I became a bit of an expert on van Est cohomology when proving some no-go theorems for Lie 2-groups in HDA6, and you can find more information about it in Section 8.5 of that paper (the section called ‘2-groups from Chern–Simons theory’). Besides the original paper of van Est, there are also useful papers by Hu and Mostow:

• S. T. Hu, Cohomology theory in topological groups, Mich. Math. J. 1 (1952), 11–59.
• G. D. Mostow, Cohomology of topological groups and solvmanifolds, Ann. Math. 73 (1961), 20–48.

Later, Brylinski invented a subtler kind of group cohomology involving ‘locally smooth’ cochains. This seems to be better than van Est cohomology for the sort of stuff we’re interested in.

For example:

There’s a very interesting nontrivial 3-cocycle

$a : G \to U(1)$

when $G$ is a compact simply-connected simple Lie group acting trivially on $U(1)$. This gives rise to an interesting and by now rather famous 2-group with $a$ as associator.

On the other hand, every smooth or even continuous cocycle of the above sort is trivial. You can see a proof in Thm. 59 of HDA6.

On the third hand, there are very interesting nontrivial locally smoothable cocycles of the above sort.

This is related to how smooth anafunctors are more important than smooth functors. Smooth anafunctors are gadgets that can be locally but not globally described as smooth functors.

Posted by: John Baez on September 24, 2008 1:18 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I’d never heard it called van Est cohomology but rather continuous group cohomology. And there’s more in van Est:
If G is a Lie group, then we have the topological cohomology H_top(G) meaning of the underlying space, we have the continuous group cohomology and we have the Lie algebra cohomology H(Lie G). If G is compact connected, there is a well known iso of two of these but van Est constructed a spectral sequence relating all 3 in general.

Posted by: jim stasheff on September 25, 2008 2:07 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

What you’re looking at is not the Postnikov tower but the Moore-Postnikov tower for a pointed space $* \to X$: what you get from generalisations of the universal cover of a space.

$\ldots \to Ninebrane(n) \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to O(n)$

is the bottom end of the Moore-Postnikov tower for the obvious pointed space $1:* \to O(n)$ (leaving out some degenerate layers where the homotopy groups are trivial)

Posted by: David Roberts on September 24, 2008 12:53 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

What do you mean by the ‘Moore–Postnikov tower of a pointed space’?

I just know about the Moore–Postnikov tower of a map

$f : X \to Y$

between pointed spaces. This is a sequence of spaces

$X \to \cdots \to X_n \to X_{n-1} \to \cdots \to X_1 \to X_0 = Y$

that interpolate between the space $X$ and the space $Y$. When $n = 0$, $X_n$ is just $Y$. Then, as $n$ increases, $X_n$ starts looking more and more like $X$ — ‘from the bottom up’.

A bit more precisely: the map $X_n \to X_{n-1}$ induces an isomorphism of homotopy groups except for the $n$th, at which point I guess it looks exactly like the map

$f_* : \pi_n(X) \to \pi_n(Y)$

Now, the Postnikov tower of a pointed space $X$ is just the Moore-Postnikov tower of the map

$X \to *$

We start out with the point $*$ and get a tower of spaces with more and more homotopy groups matching those of $X$.

I’m hoping that what you’re calling the ‘Moore–Postnikov tower of a pointed space’ is what I’d call the Moore–Postnikov tower of the map

$* \to X$

Here we start out with $X$ and get a tower of spaces with more and more homotopy groups matching those of a point. That seems to fit what you’re saying.

Alas, I haven’t figured out how this Moore–Postnikov stuff fits with what I was calling cokilling cohomology groups. I think they agree in the special case of going from $Spin(n)$ to $String(n)$, but I’m not sure how generally they agree.

In other words: can we always build the successive stages of what I’m calling the Moore–Postnikov tower of the map

$* \to X$

by successively cokilling the cohomology groups of $X$, from the bottom up?

Posted by: John Baez on September 24, 2008 3:52 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

What I wanted to say was “it’s the Whitehead tower”, but it eluded me, so I resorted to a waffling about Moore-Postnikov, which I thought was equivalent. You are correct in your guess: I meant the tower of the map $* \to X$ for a pointed space $X$.

I should check what I’ve written before.

Posted by: David Roberts on September 25, 2008 4:11 AM | Permalink | Reply to this
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Excerpt: Short essays by Beno Eckmann
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Re: Group Cocycles and Simplices

Hi all,

first of all thank you for you comments on my problem! In fact, what I have in mind is a construction of higher connected covers of topological groups, so in particular thanks for your explanations on the difference between the various possibilities to interpret this! However, I would like to take Jim’s point of view that the $n$-connected cover of an $(n-1)$-connected space $X$ is a fibration $\tilde{X}\to X$ with $\tilde{X}$ $n$-connected and $\pi_{n+k}(\tilde{X})= \pi_{n+k}(X)$.

I am quite sure that the cocycle is universal (for discrete abelian groups, just as in the case for the simply connected cover). From this it follows that cocycle in fact corresponds to the canonical element in

(1)$[G,K(\pi_n(G,n)]\cong [BG,K(\pi_n(G,n+1)]\cong H^{n+1}_{group}(G,\pi_{n}(G)),$

which you get from the map $G\to K(\pi_n(G,n)$, realising the identity on $\pi_n(G)$.

The natural object that the higher cocycles lead to are n-groups, with an associator (respectively, pentagonator, hexagonator…) given by the corresponding cocycle. For $n=2$ this yields a 2-group, which is constructed from the crossed module $\tau:\pi_2(G)\to G$ (with $\tau$ trivial) and an associator given by $(g,h,k)\mapsto(\Theta_2(g,h,k),ghk)$ for $\Theta_2$ the god given cocycle. I hope that this carries the essential structure of a $2$-fold connected cover of $G$, although the associator is not continuous (but this could be remedied by passing to some kind of multiplicative Bundelgerbe as defined in here and there).

Posted by: Christoph Wockel on September 30, 2008 1:24 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

for n > 2, calling this an group is true
but in a superficial and misleading sense.

I think it’s much more relevant that this gives 2-stages, i.e. a fibration and not an iterated fibration

Posted by: jim stasheff on October 1, 2008 12:34 AM | Permalink | Reply to this
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Re: Group Cocycles and Simplices

Is that at all related to $C \otimes_{GrayCrans} D$?

Maybe I can say a bit about that.

It’s clear that that last coend formula you had written down involves Day convolution; here we are using the tensor on $\Delta$ given by ordinal sum to induce the convolution product. But one annoying sticking point in this simplicial context is that dimensions “don’t quite add up” in the way one would expect from the Crans-Gray tensor product. The object $[p]$ in $\Delta$ has “dimension” $p-1$; that is, the affine simplex = convex hull of $p$ points in general position has dimension $p-1$. So the tensor product $[p] \otimes [q] = [p+q]$ would have dimension $p + q - 1$: one greater than the sum of the dimensions of the inputs. Whereas dimensions add “normally” for the Crans-Gray tensor product.

Of course, Crans works not with simplicial sets but with cubical sets, with a Day convolution tensor product induced from tensor products of cubes, and where the dimensions add normally. Here, we have a nerve functor

$N: \omega-Cat \to Set^{\Box^{op}}$

induced from a functor

$O: \Box \to \omega-Cat.$

As you know, this is part of a standard “Kan yoga” where the nerve $N$ has a left adjoint $F$, given by a weighted colimit or tensor product

$F X = X \otimes_{\Box} O$

which one can express more explicitly in terms of coends. In fact, the last coend formula you wrote down is of the form

$F(N C \otimes_{Day} N D)$

(replacing now $\Delta$ with $\Box$). This isn’t quite the formula for the Crans-Gray tensor product, but it is true that there is a coequalizer

$F(N C \otimes_{Day} N D) \to C \otimes_{CransGray} D$

for $\omega$-categories $C$, $D$.

To understand how this works, just bear in mind that the situation is supposed to be analogous to the way in which one can use the adjunction

$F \dashv U: Ab \to Set$

between sets and abelian groups, to define the tensor product of abelian groups: it’s defined as a certain coequalizer of the form

$F(U A \times U B) \to A \otimes B.$

To make a long story short, in order for all this to work, one needs for the adjunction $F \dashv U$ to be monoidal. And it’s the same deal here: it turns out that

$N: \omega-Cat \to Set^{\Box^{op}}$

is monadic ($\omega$-Cat is equivalent to the category of algebras for the monad $N F$), as in the abelian groups case, and the monad $N F$ is monoidal. From there you’ll probably be able to work out the correct Crans-Gray tensor product just by following the analogy.

Posted by: Todd Trimble on October 29, 2008 7:10 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Todd,

thanks for your message. I know that I am a bad student of yours. I should already know this stuff much better from our previous conversations about related things. I hope you can bear with me.

For $C$ a monoidal $V$-enriched category, presheaves $F,G : C^{op} \to V$ inherit the Day convolution tensor product defined as $(F \star G)(-) = \int^{c,d \in C} F(c)\otimes G(c) \otimes hom(-, c \otimes d) \,.$

This generalizes the familiar convolution product of functions which we recover by taking $C$ to be a discrete monoidal category, i.e. a monoidal set, i.e. a monoid, and regard for instance $F,G : C^{op} \to Set$ as categorified $\mathbb{N}$-valued functions on this monoid, in which case their Day convolution product is the ordinary convolution product of (categorified) $\mathbb{N}$-valued functions $(F \star G)(e) = \int^{c,d} F(c) \times G(d) \times hom_C(e,c \cdot d) = \oplus_{c\cdot d = e} F(c)\times G(d) \,.$

The presheaf $I$ constant on the tensor unit $I$ in $V$ is indeed the tensor unit under the Day convolution product since — let me see \begin{aligned} (F \star I)(e) &= \int^{c,d \in C} F(c)\otimes hom(e, c \otimes d) \\ &= \int^{c\in C} F(c)\otimes hom(e, c ) \\ &= F(e) \end{aligned} \,, I suppose, where about the step from the first to the second line I feel a bit shaky, while the step from the second to the third line is Yondeda reduction as you also explained here.

Okay, now with $C = \Box$ the cubical category and $V = Set$, we have underlying any $\omega$-categories $C,D$ their cubical nerves $N C = Hom( O([-]), C) : \Box^{op} \to Set$ which inherit the Day convolution product from the standard monoidal product on $\Box$ and the Crans-Gray tensor product is supposed to be a coequalizer from the free $\omega$-category on that, which is \begin{aligned} F(N C \star N D) &= \int^n (N C \star N D)([n])\cdot O([n]) \\ &= \int^{n,p,q} Hom(O([p]),C)\times Hom(O([q]),D) \times Hom([n],\underbrace{[p]\otimes [q]}_{[p+q]})\cdot O([n]) \\ &\simeq \int^{p,q} Hom(O([p]),C)\times Hom(O([q]),D) \cdot O([p+q]) \end{aligned} \,, where in the last step I did the coend over $n$ by Yoneda.

Now I need to figure out which two morphisms into this $F(N C \star N D)$ are coequalized by $C \otimes_{CransGray} D$.

Hm, so I am supposed to use that $N \circ F$ is monoidal, so that for instance $F \circ N \circ F(N C \star N D) \simeq F (N \circ F \circ N (C) \star N \circ F \circ N (C) ) \,.$ Hm, so I should use the counit in two places…

Posted by: Urs Schreiber on October 29, 2008 10:21 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Perhaps a word of caution is needed here to any passers-by of this café. It seems that here Todd is using MacLane’s categorical convention for the objects of $\Delta$, so for him $[p] = \{1,2, ... , p\}$ with the usual order.

On the other hand within sources on simplicial homotopy theory, it is more usual to write $[p] = \{0,1,2, ... , p\}$ (obvious order) and then $[p]\oplus[q]$ looks like $[p + q + 1]$, but beware it is a functor of two variables which sometimes further confuses people. Thus for simplicial homotopy $[p]$ does have the right dimension.

In the past, I have wasted quite a lot of time when reading sources on this sort of thing because I did not verify which indexing was being used. Once that is sorted the calculations are fairly routine, but clearly can get confusing if you don’t check up first!

Posted by: Tim Porter on October 29, 2008 11:12 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I wrote #:

Now I need to figure out […]

I should do the toy case of abelian groups first, as Todd suggested.

So let $C = \mathbf{B}\mathbb{Z}$ be the category with a single object and $\mathbb{Z}$ worth or morphisms. This is monoidal (it’s a 2-group even) with the standard group operation on the morphisms.

An abelian group $A$ is in particular a $\mathbb{Z}$-module hence a presheaf on $C$ $U(A) : C^{op} \to Sets \,.$

So for $A$, $B$ two abelian groups with underlying presheaves $U A$ and $U B$ with underlying sets which I’ll also write $U A := U A(\bullet)$ and $U B := U B(\bullet)$, their Day convolution product $(U A \star U B)(-) = \int^{\bullet,\bullet \in \mathbf{B}\mathbb{Z}} U A(\bullet) \times U A(\bullet) \times hom_{\mathbf{B}\mathbb{Z}}(-,\bullet)$ is the universal co-extraordinary family in that for any morphism $\bullet \stackrel{k}{\to} \bullet$ in $C = \mathbf{B}\mathbb{Z}$ we have the cone $\array{ U A \times U B \times \mathbb{Z} &\stackrel{(k\cdot-)\times Id \times (-+k)}{\to}& U A \times U B \times \mathbb{Z} \\ \downarrow^{Id \times (k \cdot -)\times (-+k)} && \downarrow \\ U A \times U B \times \mathbb{Z} &\to& \int^{\bullet,\bullet} U A(\bullet) \times U B(\bullet) \times hom_{\mathbf{B}\mathbb{Z}}(\bullet,\bullet) } \,.$ That should mean that $(U A \star U B)(\bullet) = (U A(\bullet) \otimes_\mathbb{Z} U B(\bullet)) \times \mathbb{Z}$.

Ah, so the coequalizer I am looking for needs to kill this factor of $\mathbb{Z}$.

Okay, I ended my last message # essentially with guessing that I am to coequalize the maps

$F \circ U \circ F (U A \star U B) \stackrel{ \epsilon \circ Id_{F(U A \star U B)} }{\to} F (U A \star U B)$ and $F \circ U \circ F (U A \star U B) \stackrel{\simeq}{\to} F( U \circ F \circ U A \star U \circ F \circ U A ) \stackrel{ F ( \epsilon \circ Id_{U A} \star \epsilon \circ Id_{U B} ) }{\to} F (U A \star U B) \,,$ where $\epsilon$ is supposed to denote the counit of the monoidal adjunction $\epsilon : F \circ U \to Id \,.$

This is becoming notationally a bit of a challenge now (for me at least), hope the above is readable.

Is the coequalizer of these two maps really $A \otimes_{\mathbb{Z}} B$? Looks not unplausible. The first map just plays around with free constructions, while the second actually knows the group structure on $A$ and $B$ but leaves the superfluous $\mathbb{Z}$ alone. So that might just drop out…

Posted by: Urs Schreiber on October 29, 2008 11:48 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I’ll respond to both of Urs’s comments here.

The presheaf I constant on the tensor unit I in V is indeed the tensor unit under the Day convolution

Not quite: the unit for convolution is more like the Dirac functional, not the constant at 1. Well, that’s sort of a silly way of putting it, but the point is that the unit is really $hom(-, I)$, not the constant. (The Yoneda embedding would be a universal monoidal functor, and so $yI = hom(-, I)$ would be the unit.) If you work through this and apply a “Yoneda reduction” to the resulting coend, it should all work out.

Regarding the second comment: as far as working out “what coequalizer” we use to construct the tensor product of abelian groups [as a warmup], let me be a little pedestrian about it. So: the tensor product $A \otimes B$ is the quotient of the free abelian group $F(U A \times U B)$, that is, the group of formal sums of the form $\sum_i a_i \otimes b_i$, where we mod out by equalities of the form

$(\sum_i a_i) \otimes (\sum_j b_j) = \sum_{i, j} a_i \otimes b_j \qquad (1)$

[We have to quotient out by the subgroup generated by differences between the two sides of the equation, i.e., consisting of sums of such differences.] The expressions $\sum_i a_i$, $\sum_j b_j$ can be thought of as values of typical elements in the free abelian groups $F U A$, $F U B$ under counit maps

$\varepsilon_A: F U A \to A \qquad \varepsilon_B: F U B \to B$

So the left side of this equation would correspond to the map

$F(U \varepsilon_A \times U \varepsilon_B): F(U F U A \times U F U B) \to F(U A \times U B) \qquad (2)$

The right side of equation (1) is where we use the monoidal structure on the monad $U F$, to take a tensor product of formal sums to a formal sum of tensor products. (And then the subgroup we quotient out by would involve sums of these formal sums.) If you follow this carefully, you see that the right side of equation (1) corresponds to taking the composite

$F(U F U A \times U F U B) \stackrel{F(\theta_{U A, U B})}{\to} F U F(U A \times U B) \stackrel{\varepsilon_{F(U A \times U B)}}{\to} F(U A \times U B) \qquad (3)$

where $\theta$ denotes the monoidal structure on the monad. Then the desired tensor product is obtained by taking the coequalizer of the maps (2) and (3).

Now just transcribe this formalism to monoidal monad $N F$, replacing $Set$ by $Set^{\Box^{op}}$, $Ab$ by $\omega$-Cat, and the monoidal product $\times$ on $Set$ by the monoidal product $\otimes_{Day}$ on $Set^{\Box^{op}}$.

Posted by: Todd Trimble on October 29, 2008 7:29 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

the unit for convolution is more like the Dirac functional

I should have known this, given that I know the unit for the ordinary convolution product of functions…

Okay, so let me fix that stupid pseudo-computation I gave. It’s really

\begin{aligned} (F \star hom(-,I))(e) &= \int^{c,d} F(c)\otimes hom(d,I) \otimes hom(e, c \otimes d) \\ &\simeq \int^{c} F(c) \otimes hom(e, c ) \\ &\simeq F(e) \end{aligned} where the first step is Yoneda-reduction in $d$, the second in $c$.

(By the way, in your explanation here you introduce Yoneda-reduction with the coend over the second argument of the hom, but then use it later on also in the first argument of the hom. In the above I am following this practice – blindly :-)

Posted by: Urs Schreiber on October 29, 2008 8:44 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

you introduce Yoneda-reduction with the coend over the second argument of the hom, but then use it later on also in the first argument of the hom

Yes, because the second variable $c$ in $\hom_C(d, c)$ can be viewed as the first variable in $\hom_{C^{op}}(c, d)$ – just replace $C$ by $C^{op}$ in the enriched Yoneda lemma! :-)

Posted by: Todd Trimble on October 29, 2008 10:48 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Thanks for explaining the toy example of abelian groups in more detail.

Hm, so struturally I had the right idea which two kinds of maps you wanted me to coequalize (not that there was much of a choice with all the hints you have given), but maybe my idea of regarding abelian groups as presheaves on $\mathbf{B}\mathbb{Z}$ wasn’t good. At least not what you intended. Is it wrong, though? I mean, those last two maps in my comment above #, involving Day convolution of functors $\mathbf{B}\mathbb{Z}^{op} \to Set$, what would their coequalizer be?

Coming back to $\omega$-categories:

I see that you effectively walked me through pages 16 onwards of Sjoerd Crans’s article #, something I had bounced off a bit hard last time I looked at it. But now I am beginning to see clearer. It’s only since somewhat recently that I broke some inner barrier concerning ends and coends.

I’ll have another related question. But I post that in a separate comment.

Posted by: Urs Schreiber on October 29, 2008 9:04 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

A presheaf $F: B\mathbb{Z} \to Set$ is just a set $X$ equipped with an automorphism $\tau: X \to X$; the functor $F$ applied to the morphism $n$ is $\tau^n: X \to X$. (You may have been thinking of $\mathbb{Z}-Mod \simeq Ab$.)

But, for purposes of discussing “toy” symmetric monoidal closed categories, $\mathbb{Z}$-sets with the tensor product given by

$(X \times Y)/(n \cdot x, y) \sim (x, n \cdot y)$

[where $n \cdot x := \tau^n(x)$] is an excellent choice. I happen to love this example; the category has very nice properties, and it is often easy to calculate in the category, making it a good test model for certain purposes.

Posted by: Todd Trimble on October 30, 2008 2:19 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

(You may have been thinking […]

Yes, stupid me. I should be concentrating better when posting.

Posted by: Urs Schreiber on October 30, 2008 2:38 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Todd,

in this context, maybe you can help me with the following:

I was thinking about how to elegantly prove what sounds obvious, that with $C$ an $\omega$-category and $a, b \in C_0$ objects of $C$, the $Hom$-$\omega$-category $C(a,b)$ is the pullback

$\array{ C(a,b) &\to& C^I \\ \downarrow && \downarrow^{dom \times codom} \\ pt &\stackrel{a \times b}{\to}& C \times C }$

I am likely to miss the obvious, but this led me to think about the following possible characterization of the $n$-globs:

Write $G_n$ for the $n$-glob, i.e. the unique $\omega$-category on the globular set which is represented by the object $[n]$ in the globular category.

So $G_0$ is the terminal category, $G_1$ is the interval category, $G_2$ the bigon, etc.

Write $\sigma_0, \tau_0 : G_0 \to G_1$ for the maps which inject the point into the left and the right end of the interval, respectively.

Then, I was thinking, we can get the $(n+1)$-glob from forming the cylinder over the $n$-glob and collapsing the top and bottom, i.e. as the pushout $\array{ G_n \sqcup G_n & \stackrel{(G_n \otimes \sigma_0) \sqcup (G_n \otimes \tau_0)}{\to} & G_n \otimes G_1 \\ \downarrow && \downarrow \\ pt \sqcup pt &\to& G_{n+1} } \,.$

Does that sound right?

It’s clearly true for low $n$. I was thinking there is a simple inductive argument from there on, but I would enjoy checking this with you.

Sorry for this trivial stuff.

Posted by: Urs Schreiber on October 29, 2008 9:31 PM | Permalink | Reply to this
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