Infinity-Groups with Specified Composition

March 3, 2008

Infinity-Groups with Specified Composition

Posted by Urs Schreiber

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I have a certain desire to do the one-two-three—-infinity thing for $n$ -groups while retaining specified composition.

What I mean is this: there is the

$\;\; \bullet$ bundle point of view

and the

$\;\; \bullet$ section point of view

on higher categories. The first one uses models where the existence of compositions of $n$ -morphisms is guaranteed, but not specified, while the second one explicitly specifies for any two higher morphisms and all possible ways to attach them the resulting composite.

In the first approach it is easy to say $\infty$ -group: “Kan complex with single 0-simplex”.

While that’s easy to say, it is in general hard to do anything with (at least for me). When we want to actually do something in concrete applications, we are often better off with having a model that has specified composites. (I discussed a concrete example for that recently in Construction of Cocycles for Chern-Simons 3-Bundles.)

Well, I might be just ignorant and prejudiced. But be that as it may, it should be an interesting question in its own right to see how far we can get with handling $\infty$ -groups in the second approach, where composites are specified.

There is little chance, with present technology, to handle in the second case $\infty$ -groups with full weakening allowed. On the other hand, entirely strict $\infty$ -groups would be easy to handle, but a bit insufficient. I want something which is as strict as possible while still capturing a “sufficient” degree of weakening.

And here is my condition on what I will consider as sufficient weakening:

The model of $\infty$ -groups must be closed in that for $G$ an $\infty$ -group also $AUT(G) := Aut(\mathbf{B} G)$ is an $\infty$ -group.

Because that’s what is needed for doing differential nonabelian cohomology.

Here $\mathbf{B} G$ denotes the one-object $\infty$ -groupoid given by $G$ .

For instance, if $G$ is an ordinary group, then $AUT(G)$ is the 2-group whose objects are the ordinary automorphisms of $G$ and whose morphisms are the inner automorphisms of $G$ .

Notice that if $G$ is a strict 2-group, then $AUT(G)$ is no longer a strict 3-group – but a Gray group, meaning that $\mathbf{B} AUT(G)$ is a Gray groupoid, a groupoid enriched over the category of 2-categories equipped with the Gray tensor product. In the language of crossed group structures, this amounts to passing from crossed complexes to crossed squares.

This is described in theorem 4.3 and 5.1 of

R. Brown, I. Icen
Homotopies and automorphisms of crossed modules of groupoids
(arXiv).

and David Roberts and myself talk about it in our article.

So, forming automorphism $(n+1)$ -groups of $n$ -groups takes one from the world of strict $n$ -groups into the weakened realm. But how far? Do we need fully weakened $\infty$ -groups to have that $AUT(G)$ is an $\infty$ -group if $G$ is? Or is there some explicit “semistrict” notion of $\infty$ -group in between, rather strict, but weak enough to allow for $AUT(G)$ ?

Here is my proposal for how to deal with that (following a similar remark I made in a comment here):

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The (rather obvious, but still noteworthy) appearance of Gray-structure on the automorphism 3-group of a strict 2-group was what alerted me when I learned of the tensor product $\otimes$ on the category $\omega Cat$ of strict globular infinity-categories (from the immensely helpful Todd Trimble of course, here) as described in

Sjoerd Crans
On combinatorial models for higher dimensional homotopies
PdD thesis, chapter 3: Pasting schemes for the monoidal biclosed structure on $\omega-Cat$
(ps-files).

This tensor product generalizes the Gray tensor product on strict 2-categories. In particular, it makes $(\omega Cat, \otimes)$ a closed and biclosed category.

So I began to wonder if an $(\omega Cat,\otimes)$ -enriched environment might be the right context in which to answer my question.

Being Gray, the automorphism 3-group of a strict 2-group corresponds in particular to a one object $(\omega Cat,\otimes)$ -enriched groupoid.

So write $\omega Grp$ for the full subcategory of $(\omega Cat,\otimes)-Cat$ whose objects are one-object groupoids, and

$\mathbf{B} : \omega Grp \hookrightarrow (\omega Cat,\otimes)-Cat$

for the obvious inclusion.

Then: isn’t it true that $\omega Grp$ is closed in that for each $\omega$ -group $G$ we have that

$AUT(G) := Aut_{(\omega Cat,\otimes)-Cat}(\mathbf{B} G)$

is itself such that

$\mathbf{B} AUT(G)$

is an $(\omega Cat,\otimes)$ -enriched one-object groupoid?

From looking at section 2 of

G. M. Kelly
Basic concepts of enriched category theory
(pdf)

I get the impression that the answer is: yes, of course. But I feel I need to think more about enrichment in general, and enrichment over $(\omega Cat,\otimes)$ in particular.

Posted at March 3, 2008 3:51 PM UTC

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Re: Infinity-Groups with Specified Composition

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Urs wrote:

In the first approach it is easy to say $\infty$ -group: “Kan complex with single 0-simplex”.

While that’s easy to say, it is in general hard to do anything with (at least for me).

It takes a while to get used to this Kan complex stuff, but it’s worth noting that homotopy theorists find Kan complexes incredibly nice to work with. So, they’ve been developing machinery for about 4 decades to do all sort of things with them. So, there’s a lot of stuff one can just look up.

For example, say you have an $\infty$ -group described as a Kan complex with just one 0-simplex. You might prefer to turn it into some sort of $\infty$ -group where multiplication is a specified operation. For this, homotopy theorists usually take the simplicial loop space of your Kan complex. This is an $A_\infty$ monoid — sort of like a simplicial monoid, but where multiplication is associative only up to coherent homotopy. But, there’s a way to massage this $A_\infty$ monoid to get an honest simplicial monoid, and then do ‘group completion’ to get an honest simplicial group. And, this simplicial group is another view of your original $\infty$ -groupoid!

(So, weakening the group laws is not really essential in this context, if you are willing to work with a big ‘puffed-up’ version of your $\infty$ -group: every $\infty$ -group is equivalent to one where associativity, the left and right unit laws and inverses hold strictly! Someday people will prove this in a globular setting; I don’t know if they have yet.)

It takes practice to use all this simplicial machinery, and it seems hard to find it all explained in one place. Our guardian angels, Peter May and James Stasheff, were responsible for a lot of the early work on this topic. May’s online book The Geometry of Iterated Loop Spaces has a lot of good stuff in it. His Simplicial Objects in Algebraic Topology is also useful. Jardine and Goerss’ Simplicial Homotopy Theory gives a good modern overview.

But, I’ve been studying this stuff for years (when not wasting my time on other things) and I still feel like I’m just a beginner. That’s why I’m perfectly willing to do a lot of stuff with 2-groups, just to test out certain ideas, and leave the $\infty$ -group generalizations for energetic youngsters like you.

I guess my main point is that if you want to study weak $\infty$ -groups, most homotopy theorists would consider it insane to use anything other than simplicial methods, since they’re so well-developed.

Posted by: John Baez on March 3, 2008 6:31 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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[…] And, this simplicial group is another view of your original $\infty$ -groupoid!

Okay, thanks!

That’s why I’m perfectly willing to do a lot of stuff with 2-groups

Yes. But I need to go to $n=7$ right now with a concrete computation. I know what I need for $n=7$ can be done nicely with Gray groups for $n=3$ . There is plenty of evidence that the $n=7$ -situation here is “dual” to the $n=3$ -situation. Whatever that means, it means that it is unplausible that what works nicely for $n=3$ necessarily needs to becomes a mess for $n=7$ .

Passing to fluffy Kan complexes in this example just doesn’t feel right. So I am trying to see if this is supposed to be pointing me to a nice semistrictification of $\infty$ -groups.

Whether or not it does: am I right that $(\omega Cat,\otimes)-Cat_{small}$ is itself $(\omega Cat,\otimes)$ -enriched?

Posted by: Urs Schreiber on March 3, 2008 8:03 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

A special example of an infinity-group: a Kan complex with single 0-simplex is the subcomplex of the singular complex Sing(X) of a pointed space (X,*) consisting of simplices such that all vertices map to *.
If we topologize this in $Maps(\Delta^n, X)$,
then we get an alternated construction for the classifying space of $\Omega X$ due, if I recall correctly, to Graeme Segal.

Is this already here somewhere on the blog or at n-Lab?

Posted by: jim stasheff on September 24, 2009 2:13 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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How are you making $(\omega Cat,\otimes)$ -Cat enriched over $(\omega Cat,\otimes)$ ? Any closed monoidal category $V$ is enriched over itself, and hence any object $X$ of $V$ has an endomorphism object $End(X)$ , which is a monoid (1-object category) in $V$ . This applies to $(\omega Cat,\otimes)$ , so any object of $\omega Cat$ has an endomorphism object in $Mon(\omega Cat)$ , and (one assumes) an automorphism object in your $\omega Grp$ . However, you still have to go up one level: the automorphisms of an $(\omega Cat,\otimes)$ -enriched category (such as an $\omega$ -group in your sense) will be not an $(\omega Cat,\otimes)$ -enriched category but an $(\omega Cat,\otimes)$ -Cat-enriched category.

In the fully strict case, we have $(\omega Cat,\times)$ -Cat = $\omega Cat$ , so you can get away without going up a level. But I don’t think that will work with the tensor product $\otimes$ . This is almost the same as my earlier objection: an $(\omega Cat,\otimes)$ -enriched category will have weak composition along 0-cells, but all its other compositions will be strict. But when you try to talk about automorphism objects, you end up needing weakness at one level further up.

Posted by: Mike Shulman on March 3, 2008 8:28 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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the automorphisms of an $(\omega Cat,\otimes)$ -enriched category (such as an $\omega$ -group in your sense) will be not an $(\omega Cat,\otimes)$ -enriched category but an $(\omega Cat,\otimes)-Cat$ -enriched category.

Oops. Thanks. My fault. Yes, I see. And yes, I see you tried to tell me before.

And indeed, the automorphisms of a 0-Cat-enriched category form a 1-Cat-enriched category with a single object.

And the automorphisms of that form a $(1Cat)-Cat = (2Cat,\otimes_{Gray})$ -enriched one object thing, as I have been saying myself above.

Okay, I need to think about this again.

But does this continue indefinitely

$(((\omega Cat,\otimes)-Cat)-Cat)-Cat)$

etc, with automorphisms of an $(\cdots((\omega Cat,\otimes)-\underbrace{Cat)-\cdots-Cat)}_{n}$ -category forming an $(\cdots((\omega Cat,\otimes)-\underbrace{Cat)-\cdots-Cat)}_{n+1}$ -enriched category?

Posted by: Urs Schreiber on March 3, 2008 9:15 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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But does this continue indefinitely[?]

I think so, in principle. However, there are issues making it precise, since I don’t know of a good tensor product on $\omega Cat$ -categories. It’s analogous to the problem of finding a good tensor product of Gray-categories, which seems to have no fully satisfactory solution.

Posted by: Mike Shulman on March 3, 2008 10:52 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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For what it’s worth: both James Dolan and I, and later Brian Day and Ross Street, tried to formalize a general concept of ‘Gray tensor product’. The idea was to start with a sufficiently nice monoidal category $C$ , construct a ‘Gray tensor product’ on $C-Cat$ , and prove this made $C-Cat$ into another sufficiently nice monoidal category. If we could do that, we could iterate this and get a theory of Gray $n$ -categories.

In both cases the idea failed in the same subtle way! $C$ being sufficiently nice did not imply that $C-Cat$ was sufficiently nice. But, the flaw took a while to show up!

Start with $C = Set$ with its usual cartesian product. Then $C$ is sufficiently nice, and $C-Cat$ works out to be $Cat$ with its usual cartesian product.

$C-Cat$ is sufficiently nice, and $(C-Cat)-Cat$ works out to be $2Cat$ with its usual Gray tensor product.

$(C-Cat)-Cat$ is sufficiently nice, and $((C-Cat)-Cat)-Cat$ works out to be $3Cat_{semistrict}$ , or what the Australians call $\mathbf{Gray}-Cat$ .

Unfortunately, $((C-Cat)-Cat)-Cat$ turns out not to be sufficiently nice! So, the crank falls off the machine at this point.

Just so you don’t repeat this mistake, the idea was to work with monoidal categories equipped with an ‘interval object’ that allows one to define ‘homotopies’, and use these to mimic the usual Gray tensor product of $Cat$ -categories, where certain squares commute ‘up to homotopy’.

If you try something like this, be very careful: there’s quicksand around here!

Posted by: John Baez on March 3, 2008 11:56 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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The progression

sets –> categories –> 2-categories –> Gray-categories –> …

kind of reminds me of the progression

real numbers –> complex numbers –> quaternions –> octonions –> …

in which you lose something at every step, until finally you’ve lost so much that it’s not worth going on.

I also confess to having spent time trying to generalize the Gray tensor product, and of course failing. I don’t suppose either you and James Dolan or Day and Street ever wrote down any of what you did? It seems a common enough trap that it would be useful having a well-explained development of exactly what goes wrong and why, to save time and effort on the parts of future would-be Gray-tensor-product-generalizers. It might also help us see whether there is something we could give up and still get a workable theory.

I’m also curious what other categories are “sufficiently nice” according to your definition. For instance, is there a Gray tensor product of topologically enriched categories? Simplicially enriched ones? Such products would be useful to have, even if it’s not possible to iterate them.

A related question that I’ve been wondering about recently: there is an explicit tensor product for Gray-categories. The resulting monoidal category is not closed, but we could still, if we wanted, use it to define a notion of “semi-strict 4-category”. Is the resulting notion too strict? Does anyone know?

Posted by: Mike Shulman on March 4, 2008 2:28 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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I also confess to having spent time trying to generalize the Gray tensor product, and of course failing. I don’t suppose either you and James Dolan or Day and Street ever wrote down any of what you did?

Street had begun writing up material on this which he called “files” (or $n$ -files), which I might be able to find in files of my own (he he), or Ross might have kept something on this. I remember the email message Dolan sent which claims that Ross’s idea wouldn’t fly, which as I recall was very brief and sketchy. (John, do you still have this? I tried looking for it myself.) Now it’s quite possible that Ross later applied the term “file” to a different but related notion (in view of a possible general method for semi-strictifying), so one might have to specify, “files, circa 1995”.

I realize these remarks aren’t terribly helpful; I just mean to suggest that much of this material may well be extant. The Crans paper you linked (Mike) has remarks on this as well; he dates the Dolan email to 1995 [which surprises me actually; it’d have to be very late 1995]. I suppose someone could email Ross and check up on this.

Posted by: Todd Trimble on March 4, 2008 3:31 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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The idea was to start with a sufficiently nice monoidal category $C$ , construct a ‘Gray tensor product’ on $C−Cat$ , and prove this made $C−Cat$ into another sufficiently nice monoidal category.

If $C$ is symmetric monoidal, then $C-Cat$ is automatically symmetric monoidal, too. But here you refer to constructing another tensor product on $C-Cat$ than this standard one, right?

As Ross Street recalls on p. 2 of his An Australian conspectus of higher categrories, starting with $C = Set$ and then forming $(\dots(C-Cat)-Cat)-\cdots-Cat)$ leads to strict $n$ -categories.

But if we’d start with $C = \omega Cat$ which already comes with a Gray tensor product, it might seem we get the best of both worlds: the automatically guaranteed existence of a symmetric monoidal tensor product on $(\dots(C-Cat)-Cat)-\cdots-Cat)$ and at the same time “Gray-ness”.

In fact, let me see if I understand this:

if $C$ is symmetric monoidal and closed, then so is $C-Cat$ (maybe up to a size issue which can be dealt with).

Is that right?

Probably it’s not, because otherwise Mike wouldn’t say

I don’t know of a good tensor product on $\omega Cat$ -categories.

But what goodness do we need? If $C$ is symmetric monoidal, then so is $C-Cat$ . Isn’t that sufficient?

Posted by: Urs Schreiber on March 4, 2008 10:52 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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if $C$ is symmetric monoidal and closed, then so is $C$ -Cat

This is true, as long as $C$ is also complete—which all the categories under consideration are, and which is also preserved under passage from $C$ to $C$ -Cat (by which I mean small $C$ -categories, to avoid size issues).

You can certainly iterate the construction $C\mapsto C$ -Cat starting with any symmetric monoidal category $C$ , including $\omega Cat$ . The problem is that the monoidal structure on $C$ -Cat induced from that of $C$ is usually not `good’, even if the monoidal structure on $C$ was good. Perhaps the confusion is that good doesn’t just mean closed; even the cartesian monoidal structure on $2 Cat$ is closed, and we know that isn’t good enough.

I don’t have a precise definition of good. If $C$ -Cat has a model structure, then it would be natural to ask that the tensor product be compatible with that model structure. Another natural thing to ask is that the corresponding internal-hom capture not just strict natural transformations, but pseudo ones in a suitable sense. This is one way to define the classical Gray tensor product: it is the tensor product corresponding to the internal-hom on $2Cat$ where $[A,B]$ is the 2-category of strict 2-functors, pseudonatural transformations, and modifications from $A$ to $B$ .

This latter condition will be violated by the monoidal structure on $C$ -Cat induced by that of $C$ whenever $C$ -categories have a notion of `pseudo’ natural transformation which is different from the `strict’ notion of $C$ -natural transformation, since the internal-hom for the induced monoidal structure is known to represent only the strict ones. When $C=Set$ , there is no nontrivial notion of pseudo, so the induced product is good. But even when $C=Cat$ , with its (good) cartesian monoidal structure, the induced monoidal structure on $C$ -Cat $= 2 Cat$ is again the cartesian product, which is not good; one needs the Gray tensor product on $2 Cat$ . Similarly, starting with the Gray tensor product on $C=2 Cat$ , the induced monoidal structure on Gray-Cat is not good. So there is no reason to expect that the monoidal structure on $\omega Cat$ -Cat induced from the Gray product on $\omega Cat$ will be good.

Put another way, “Grayness” is not a binary property; a tensor product can be “Gray” at one level and not at another. To be `good’, a tensor product should be Gray at all levels, but the induced product on $C$ -Cat will never be Gray at the bottom level, although it can inherit Grayness at higher levels from the product on $C$ .

Posted by: Mike Shulman on March 4, 2008 5:22 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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Thanks, Mike, for this response. I was getting a little worried. What you say helps clarifying the situation for me a lot.

You can certainly iterate the construction $C \mapsto C-Cat$ starting with any symmetric monoidal category C, including $\omega Cat$ .

Okay.

The problem is that the monoidal structure on $C-Cat$ induced from that of $C$ is usually not ‘good’, even if the monoidal structure on $C$ was good. Perhaps the confusion is that good doesn’t just mean closed;

Yes! I was left a little puzzled what good was supposed to mean.

even the cartesian monoidal structure on 2Cat is closed, and we know that isn’t good enough.

[…]

Another natural thing to ask is that the corresponding internal-hom capture not just strict natural transformations, but pseudo ones in a suitable sense.

[…]

This latter condition will be violated by the monoidal structure on $C-Cat$ induced by that of $C$

Ah, I get it. Thanks a lot.

To be ‘good’, a tensor product should be Gray at all levels, but the induced product on $C-Cat$ will never be Gray at the bottom level, although it can inherit Grayness at higher levels from the product on $C$ .

Thanks. This is something I should think about in a concrete example, to see if and how it works for the special applications I have.

Thanks again.

Posted by: Urs Schreiber on March 4, 2008 5:37 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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I have another potentially stupid question:

Starting with $V = Set$ , we have canonical inclusions

$(\cdots(V-\underbrace{Cat)-Cat)-\cdots)-Cat}_n \hookrightarrow (\cdots(V-\underbrace{Cat)-Cat)-\cdots)-Cat}_{n+1}$

namely

$n Cat \hookrightarrow (n+1)Cat$

in this case.

Under which conditions on $V$ do we have such inclusions?

Could it be that it is a rather peculiar coincidence that this works for $V = Set$ , essentially due to the fact that a $V$ -category happens to have a set of objects?

Back to my original motivation:

One expects, for a given model of $n$ -groups, that for $G$ an $n$ -group and $AUT(G)$ its automorphism $(n+1)$ -group, there is a canonical inclusion

$Ad : G \hookrightarrow AUT(G)$

sending $k$ -morphisms in $G$ to “inner” automorphisms obtained from conjugating with these.

Now, back to the case where I can think of $G$ as a one-object $(\omega Cat, \otimes_{Gray})$ -groupoid, such that $AUT(G)$ will be a one object $(\omega Cat, \otimes_{Gray})-Cat$ -groupoid (I think we agreed that this is how it works, possibly up to the fact that we haven’t specified what “groupoid” here is supposed to mean). Then: can I expect to have a canonical inclusion

$G \hookrightarrow AUT(G)$

Posted by: Urs Schreiber on March 5, 2008 5:59 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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Could it be that it is a rather peculiar coincidence that this works for $V=Set$ , essentially due to the fact that a $V$ -category happens to have a set of objects?

That would be my inclination. At least, I don’t see any way to do it for general $V$ . For $n=0$ , you’d need a way to regard an object of $V$ as a $V$ -category, and I don’t know any way to do that in general. Think of $V=Top$ , for instance.

On the other hand, it does work for $V=(\omega Cat,\times)$ , since $(\omega Cat,\times)$ -Cat $= \omega Cat$ . And it works for $V=(2Cat,\otimes_{Gray})$ , since I can regard a 2-category as a strict 3-category, and thereby as a Gray-category where the interchange isomorphisms happen to be strict. This seems to be because I have a canonical transformation $\otimes_{Gray} \to \times$ for 2-categories, so any strict 3-category becomes a Gray-category in a canonical way. I wouldn’t be surprised if the same were true for $\omega Cat$ , so that any strict $\omega$ -category becomes a $(\omega Cat,\otimes_{Gray})$ -category in a canonical way.

However, I’ll be surprised if there are very many interesting one-object $(\omega Cat,\otimes_{Gray})$ -groupoids, since that notion seems too strict at most levels.

Posted by: Mike Shulman on March 5, 2008 9:38 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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you’d need a way to regard an object of $V$ as a $V$ -category

Yes, that reminds me of Walters’ “ $W$ -categories” (as described and referred to by that name on p. 9 of the Australian conspectus): these have objects coming from the category $W$ .

But I am not claiming that I see that this vague observation would lead to anything useful…

However, I’ll be surprised if there are very many interesting one-object $(\omega Cat, \otimes_{Gray})$ -groupoids, since that notion seems too strict at most levels.

Well, all automorphism groups $AUT(G)$ of strict $\omega$ -groups $G$ are examples (as it seems you agreed?).

And these are in fact my motivating examples for the entire discussion here:

I would be happy to work entirely within $\omega$ -groups, were it not for the fact that I need to be able to form $AUT(G)$ for a given $G$ . But, at least without further work, forming $AUT(\cdot)$ means leaving the context of $\omega$ -groups and entering $(\omega Cat, \otimes_{Gray})$ -groups.

So I was trying to identitfy the strictest possible $\infty$ -context that still allows me to form $AUT(\cdot)$ .

But if I don’t also get an inclusion

$G \hookrightarrow AUT(G)$

with all that, I can discard this idea anyway.

Namely, what I really want in the end is an $\infty$ -version of the fact that for $G$ a strict 2-group, we get sequence

$\array{ G &\hookrightarrow& INN(G) &\to& \mathbf{B} G \\ && \subset \\ && AUT(G) }$

the realization of whose nerve is a model for the universal $G$ -2-bundle.

Posted by: Urs Schreiber on March 5, 2008 10:13 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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Yes, that reminds me of me of Walters’ “ $W$ -categories”

I think that’s different. There $W$ is a bicategory, and the bicategory corresponding to our monoidal category $V$ is shifted up a level. So an object of $V$ is really a 1-cell of $W$ , not an object of $W$ .

Well, all automorphism groups AUT(G) of strict ω-groups G are examples (as it seems you agreed?).

If “strict ω-group” means “strict $\omega$ -category with one object” (+ invertibility), then I do agree. I don’t think I’ve ever seen any examples of such things, though (except for, say, the abelian case in which they reduce to chain complexes), so I’d be interested to see yours.

Note that (I think) what you’ll be getting with this AUT consists of strict $\omega$ -functors $G\to G$ , together with pseudo transformations and higher cells between them, since that’s what Gray homs look like. If you want to include “pseudo $\omega$ -functors” too, you could probably replace your $G$ with a cofibrant one first.

Anyway, as I said, it seems to me that regarding an $\omega$ -category as a $(\omega Cat,\times)$ -category, and thereby as a $(\omega Cat,\otimes)$ -category where all the “Grayness” is trivial, might work for what you are describing. But I couldn’t say for sure without looking in more detail at the Gray $\otimes$ for $\omega$ -categories.

Posted by: Mike Shulman on March 6, 2008 2:02 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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If “strict $\omega$ -group” means “strict $\omega$ -category with one object” (+ invertibility),

Yes!

then I do agree. I don’t think I’ve ever seen any examples of such things,

(Maybe I misunderstand what you mean, but…) I bet you have: for $n=2$ (discussed in HDA V) where they are equivalent to crossed modules of ordinary groups, and a bit for $n=3$ we have been talking a lot about these things here at the Café.

Generally, $\omega$ -groups are equivalent to crossed complexes of groups. These are almost like complexes of abelian groups, only that the stuff in degree 1 and 2 need not be abelian.

Crossed complexes (and their relation to $\omega$ -groups) have been studied in great detail over the years by Ronnie Brown and collaborators. He has lots of information and links on that at his website:

Ronnie Brown, Nonabelian algebraic topology

There he also presents that diagram giving the big picture of his work:

The “cubical $\omega$ -groupoids with connection” appearing in the bottom right corner here are equivalent to globular $\omega$ -groupoids.

though (except for, say, the abelian case in which they reduce to chain complexes),

In a way, crossed complexes are like chain complexes of abelian groups, but taking care of the fact that the first fundamental group of a space need not be abelian.

so I’d be interested to see your [example].

As described here, the strict String 2-group can be thought of as arising as the strict fundamental 2-groupoid

$\mathbf{B} String(G) = \Pi_2(X_{CE(g_{\mu_3})})$

of the space $X_{CE(g_{\mu_3})}$ which is obtained from the Lie 2-algebra coming from the semisimple Lie algebra $g$ equipped with its canonical Lie algebra 3-cocycle $\mu_3$ (originally introduced in HDA VI).

What we are interested in, for reasons indicated in section 3 of $L_\infty$ -connections, is the strict Lie 6-group

$Fivebrane(G)$

which is similarly obtained from the Lie algebra $g$ equipped with a 7-cocycle $\mu_7$ as the fundamental 6-groupoid of the corresponding smooth space

$\mathbf{B} Fivebrane(G) = \Pi_6(X_{CE(g_{\mu_7})}) \,.$

In order to describe $Fivebrane(G)$ 6-bundles with connection, we need to refine a cocycle for a $Fivebrane(G)$ -6-bundle

$g : Y^\bullet \to \mathbf{B} Fivebrane(G)$

(where $Y^\bullet$ denotes the strict 6-groupoid coming from a good cover $Y \to X$ of the base space $X$ of the 6-bundle)

to a differential nonabelian cocycle

$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} Fivebrane(G) \\ \downarrow && \downarrow \\ \Pi_7(Y^\bullet) &\stackrel{tra}{\to}& \mathbf{B} INN(Fivebrane(G)) } \,,$

where $\Pi_7(Y^\bullet)$ denotes the weak pushout of

$\array{ \Pi_7(Y^{[2]}) &\to& \Pi_7(Y) \\ \downarrow \\ \Pi_7(Y) }$

which is the strict 7-groupoid generated from 7-paths in $Y$ and “jumps” in the fibers of $Y$ , modulo some in principle obvious relations.

Generally, I write here $\Pi_n(S)$ for the version of the fundamental smooth $n$ -groupoid of a smooth space $S$ whose $k \lt n$ -morphisms are thin-homotopy classes of smooth globular $k$ -paths in $S$ , and whose $n$ -morphisms are ordinary homotopy classes of smooth globular 7-paths.

Anyway, to even write down the above diagram, one needs

$INN(Fivebrane(G)) \subset AUT(Fivebrane(G)) \,.$

Of course I could work entirely within a non-strict model of $n$ -groupoids and this wouldn’t be a problem at all. But, as I said, for many computations it is very useful to have a strict model, so I was wondering how far I could get with that, while still considering $AUT(\cdot)$ .

Posted by: Urs Schreiber on March 6, 2008 11:17 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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Of course, I am familiar with the $n=2$ case, but I was thinking about $n>2$ where the difference between strict and weak becomes contentful. I haven’t been reading all the posts at the cafe, so I must have missed the ones talking about strict 3-groups; could you provide some links?

I’m not sure that I’m ready to dive into $L_\infty$ -connections just yet, and a brief glance at section 3 didn’t answer my main question, so let me ask it explicitly in case you can give a short conceptual answer. Namely, why is $FiveBrane(G)$ a strict 6-group? It seems that this is saying something about the vanishing of various higher homotopy invariants of the space $X_{CE(g_{\mu_7})}$ ; is there some conceptual reason for that?

Posted by: Mike Shulman on March 7, 2008 2:39 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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Of course, I am familiar with the $n=2$ case,

Okay, sorry.

I haven’t been reading all the posts at the cafe, so I must have missed the ones talking about strict 3-groups; could you provide some links?

I mostly talked about Gray groups in general, namely those arising from weak cokernels of 2-groups (here (inner automo. 3-group) and here (weak cokernels of 2-groups)), but in many special cases of interest, notably that one discussed here (Chern-Simons cocycles) (also slide 523) these become strict 3-groups.

For instance there is the strict 2-group $String(G)$ which comes from a crossed module of the form

$(\hat \Omega G \to P G)$

where the right group is paths in $G$ , the left one centrally extended loops in $G$ .

This sits in a short exact sequence

$1 \to \mathbf{B} U(1) \to String(G) \to G \to 1$

(for instance section 4.2 here)

and we can ask what the obstruction is to lifting a $G$ -cocycle to a $String(G)$ -2-cocycle.

To do so we notice that we can lift every $G$ -cocycle to the a cocycle for the weak cokernel

$(\mathbf{B} U(1) \to String(G)) \,.$

This is a strict 3-group (it happens to be strict here, since $\mathbf{B} U(1)$ is so boring) coming from a crossed complex of groups

$(U(1) \hookrightarrow \hat \Omega G \to P G) \,.$

Then projecting out from this 2-cocycle the 3-coycle obtained under

$(U(1) \hookrightarrow \hat \Omega G \to P G) \to (U(1) \to 1 \to 1)$

yields the 3-cocycle obstructing the lift (a “Chern-Simons 3-cocycle”).

That’s maybe the most interesting example that I have talked about. But notice that one gets lots of examples of non-trivial crossed complexes from (filtered) spaces, by forming their “fundamental $\omega$ -groupoids”.

I guess the best review for that is

Ronnie Brown, Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems.

I’m not sure that I’m ready to dive into $L_\infty$ -connections

No need. You asked for the example I had in mind, and I tried to provide the context. But the only relevant point for the discussion here is that $L_\infty$ -algebras are a source of spaces, and that spaces are a source for $\omega$ -groupoids.

a brief glance at section 3 didn’t answer my main question

Oh, it wasn’t supposed to answer your question! It certainly does not and is not intended to. I just mentioned it to indicate where that example you asked me to describe arises from.

why is $FiveBrane(G)$ a strict 6-group?

I tried to say that in the second but last paragraph of my above comment: in the present setup, it’s strict by construction.

I am claiming that for any smooth space $X$ and any integer $n$ , we can form a strict globular $n$ -groupoid $\Pi_n(X)$ which has thin-homotopy classes of globular $k$ -paths as $k$ -morphisms for all $k \lt n$ and has ordinary homotopy classes of globular $n$ -paths as $n$ -morphisms.

This is just a slight variant of the fundamental $\omega$ -groupoid of a space that is described in

Ronnie Brown, A new higher homotopy groupoid: the fundamental globular $\omega$ -groupoid of a filtered space.

For low $n$ the $\Pi_n(X)$ that I have in mind is described in more detail in section 2.1 arXiv:0802.0663 (well, there it’s thin homotopy all the way up, so its called $P_2(X)$ instead of $\Pi_2(X)$ , but it’s trivial to get from there to $\Pi_2(X)$ ).

It seems that this is saying something about the vanishing of various higher homotopy invariants of the space $X_{CE(g_{\mu_7})}$ ; is there some conceptual reason for that?

The thing is strict by construction. The question is how much information about the original space it retains. Probably the best general answer to that is the list of theorems summarized in Ronnie Brown’s diagram which I linked to above.

Specifically, in the special example of the String 2-group I mentioned, there is the space

$X_{CE(g_{\mu_3})}$

and we can wonder how its various flavours of path $n$ -groupoids are related.

In math/0603563 André Henriques forms a weak path $\infty$ -groupoid from it (essentially just the Kan complex of singular simplices in that space, but then also truncated at some point).

In that entry on Chern-Simons cocycles I mentioned #, I talk about the strict path 2-groupoid of that space. There I conclude that this strict path 2-groupoid is essentially $\mathbf{B} String(G)$ , with $String(G)$ the strict version of the String 2-group.

And it is known that both version of the path groupoid of $X_{CE(g_{\mu_3})}$ have equivalent realizations.

(These two approaches are mentioned on slides 152-7).

But I wish I would understand this relation between weak and strict fundamental path $\infty$ -groupoids of spaces better and more generally. I’d be glad if we discuss this further.

Posted by: Urs Schreiber on March 7, 2008 11:02 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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The thing is strict by construction. The question is how much information about the original space it retains.

Ah, okay. That is what I was missing. Sorry for being dense.

Posted by: Mike Shulman on March 7, 2008 6:42 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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Sorry

No problem at all, I was probably being unclear. I very much enjoyed this exchange!

Posted by: Urs Schreiber on March 7, 2008 7:05 PM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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I wrote:

But I wish I would understand this relation between weak and strict fundamental path $\infty$ -groupoids of spaces better and more generally. I’d be glad if we discuss this further.

In particular, this here is puzzling me:

let $g$ be an ordinary Lie algebra. It sits inside its Lie 2-algebra of inner derivations

$\array{ g \\ \downarrow \\ inn(g) } \,,$

passing to the dual CE-algebras gives the surjection of the Weil algebra onto the CE-algebra

$\array{ CE(g) \\ \uparrow \\ CE(inn(g)) &=& W(g) }$

then forming spaces from that (along these lines)

$S \left( \array{ CE(g) \\ \uparrow \\ CE(inn(g)) &=& W(g) } \right) = \array{ X_{CE(g)} \\ \downarrow \\ X_{W(g)} }$

and then forming the fundamental strict 2-groipoids of these yields

$\array{ \mathbf{B} G \\ \downarrow \\ \mathbf{B} INN(G) &=& \mathbf{B} (G // G) } \,,$

where $G$ is the simply connected Lie group integrating $g$ .

(For the top item this is standard, for the bottom item I think this is true, but I haven’t written up a full proof.)

So this yields two different ways to arrive at this universal $G$ -bundle thing: either form inner derivations at the Lie algebra level and then integrate, or first integrate and then form inner automorphisms.

But it’s not quite as simple as we move up in dimension:

starting with a strict 2-group $G_2$ coming from a strict Lie 2-algebra $g_2$ , we know that $INN(G_2)$ is not a strict 3-group in general, but a Gray group. On the other hand, that procedure of forming strict fundamental $n$ -groupoids will integrate

$\array{ g_2 \\ \downarrow \\ inn(g_2) }$

to some morphism of strict Lie 3-groups,

$\array{ \mathbf{B} G_2 \\ \downarrow \\ \Pi_3(X_{W(g_2)}) } \,.$

There should certainly be an equivalence

$\mathbf{B} INN(G_2) \simeq \Pi_3(X_{W(g_2)}) \,,$

simply because $W(g_2)$ and $INN(G_2)$ are both “contractible”, but I need to eventually better understand how that goes along with the rest of the structure and what’s really going on here.

(I have recently talked about that point with David Roberts here.)

Posted by: Urs Schreiber on March 7, 2008 11:36 AM | Permalink | Reply to this

Re: Infinity-Groups with Specified Composition

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Given that $n\mathrm{Cat}$ is $(n-1)\mathrm{Cat}-\mathrm{Cat}$ , it might make sense to count $(\omega+1)\mathrm{Cat} := (\omega\mathrm{Cat},\otimes_{Gray})-\mathrm{Cat} \,.$ Then we can climb the ladder again $(\omega+2)\mathrm{Cat}, (\omega+3)\mathrm{Cat}, \dots \,.$

Do we similarly get to $2\omega\mathrm{Cat}$ this way?

And then?

Posted by: Urs Schreiber on April 9, 2008 12:48 AM | Permalink | Reply to this

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March 3, 2008