The n-Category Café

That article looks like a very self-contained didatic book, with subjects properly listed on the index and solved exercises at the end.

Just to be very sure that there is no misunderstanding I want to emphasize again that this is work in progress, as one can see from all the internal comments inside the document and from the inclomplete sections.

As far as self-contained goes: for the moment we have that “glossary” section at the end. This arose from discussion who the intended audience of this stuff should be. One important motivation for the entire development is to explain structures and phenomena that arise in quantum field and string theory. But many of the tools we need tend to be (of course there are exceptions) unknown among those who currently appreciate the relavance of the applications.

And conversely. We also have a “glossary” section (very brief at the moment) which goes the other way round and tries to (that’s at least the original intention, in its present state this may well fail at the moment) point out to those familiar with the abstract nonsense that there are interesting applications out there which are waiting to get treated with this abstract stuff.

So that’s where this glossary comes from. It is not the kind of thing that is likely to survive once one thinks about formal publication, but for the time being I think it is good to keep it there. (Of course it would be even better to further expand on it. Volunteers should email me ;-).

and solved exercises at the end.

The second part is titled “Examples and applications” because two things are discussed here, after the development (or what exists of that) of the general theory in the previous part:

i) familiar examples are shown to be reproduced. You might call this “solved exercises”. And ii): the machinery is applied to new examples, in particular to those lifting problems through the sequence of $\omega$-groups $Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to O(n)$. Well, perhaps this is an just exercise, too. It certainly is in a way in that everything here is supposed to follow straightforwardly by just turning the crank on the machinery developed before.

Bruce is wondering # about the context of $\omega$-groupoids (strict $\infty$-categories with strict inverses).

Here are some remarks:

a) just to make sure, there is not a claim here, at the moment, to the extent “$\infty$-Lie integration to strict smooth $\infty$-groupoids is/is not more general/less general in some sense than integration to weak $\infty$-groupoids”.

What’s happening is that one observes that the Sullivan/Getzler method can be understood as first forming the classifying space of flat $L_\infty$-algebra valued forms, and then probing that using a weak path $\infty$-groupoid. The information about the $L_\infty$-algebra is really all in that classifying space, and we just need to see how to extract it. The observation is that for every space we also have its strict (smooth) fundamental $\infty$-groupoid, and that in the examples we have worked out (in low dimensions mostly, but not only) forming the strict smooth $\infty$-groupoid of that space yields the strictification of the weak $\infty$-groupoid.

b) maybe one should keep in mind that $L_\infty$-algebras are also not the weakest possible Lie structure: we knew from the get-go (as for instance remarked in John and Alissa’s HDA VI) that there should be weak $L_\infty$-algebras which are more general in that they are skew-symmetric only up to higher coherent equivalence. Roytenberg recently worked that out for $n=2$, as you know.

So maybe a good question to ask is: what would these weak $L_\infty$-algebras be like, how would we form their classifying spaces, and how could we probe these? What would weak $L_\infty$-algebras integrate to?

You may remember that I made comments on this question a couple of times here:

Let’s ask where the skew-symmetry of a Lie algebra bracket comes from: I think it makes sense to say that it comes from the fact that a Lie group has strict inverses! Because the Lie bracket is the differential of the group commutator $\frac{d}{dt}|_{t=0} g(t) h(t) g(t)^{-1} h(t)^{-1}$ and writing this out with exponential maps and looking at the familiar formulas from a higher perspective, one sees that the skew symmetry of the bracket reflects the fact that $g(t)^{-1}$ really is the inverse of $g(t)$, not up to something.

So, I am thinking that: maybe it makes sense to integrate $L_\infty$-algebras to strict $\infty$-groupoids, because weak $\infty$-groupoids should be the integrated versions of weak $L_\infty$-algebras!

This is just an idea, I don’t have any precise statement here. But maybe it is an idea worth of keeping in the present context.

c) Or maybe not. Maybe forming the strict path $\infty$-groupoid of the classifying space of flat $L_\infty$-algebras does lose information. As I mentioned elsewhere, I currently don’t have a complete idea (though some partial ideas) of what should be the “third $\infty$-Lie theorem”. Instead, $\omega$-categories and $\omega$-groupoids currently serve for me the purpose that they are tractable and that we can use them in practice to attack the questions that ought to be attacked. I keep in mind the option of eventually weakening the entire setup and allowing weak $\infty$-categories. But for the moment sticking with the strict ones allows to avoid getting lost in just foundational and technical questions and to instead make some progress on applications.

Sjoerd Crans somewhere in his articles says a wise word along the lines (just roughly paraphrasing from memory): before trying to tackle weak $\infty$-categories let’s make sure we have completely mastered the strict ones.

He is speaking of the general theory. But the same kind of comment should apply to applications of $\infty$-categories: let’s see how many applications can be handled already with strict $\infty$-categories. We can still eventually try to weaken everyhting if we see we are getting stuck otherwise.

(a) When you form the smooth space $K(G)$ corresponding to a smooth $\omega$-groupoid $G$, how come it isn’t just the smooth space of 0-cells, ie. $K(G) = G_0$, as opposed to the construction $K(G) = Hom(\Pi_\omega(-,)G)$. I guess my construction is way off, because for a group it would just give you a smooth space consisting of a single point. But it shows I have a bit of confusion here.

Here I am not sure what the question really is! As you say, one needs to “probe” all higher cells of $G$.

More abstractly, this is what happens to be the right adjoint to $\Pi_\omega : Spaces \to \omega Categories(Spaces)$. See the proof of prop. 4.16, p. 66.

(b) Have you shown that integrating an $L_\infty$-algebra to a smooth $\omega$-groupoid and then differentiating it again indeed reproduces the original $L_\infty$-algebra?

Only for special cases in low dimensions so far.

It turns out that there, up to the case of strict Lie 2-algebras and Lie 2-groups, this is really just a direct corollary, just a rereading of the theorem in [SWII], which says that $$Hom(P_2(U), \mathbf{B}G) = \Omega^\bullet(U,g)$$ for $G$ a Lie 2-group and $g$ its strict Lie 2-algebra, which implies for the smooth fundamental 2-groupoid $$Hom(\Pi_2(U), \mathbf{B}G) = \Omega^\bullet_{flat}(U,g) \,.$$ So this says that as sheaves we have $$Hom(Pi_2(-),\mathbf{B}G) = S(CE(g))$$ !

This is currently mentioned, a bit briefly in section 5.3, p. 89.

But what I don’t have, and what nobody had last time I checked (but let’s see what Pavol &Scaon;evera will say in his talk in Lausanne next week!) is a “third $\infty$-Lie theorem” which tells us what exactly happens when we first integrate and then differentiate.

I was thinking it is a good first step to realize that the integration and differentiation procedure is the concatenation of two adjunctions. I regard that as a first, albeit incomplete, step at “third $\infty$-Lie”.

Notice that in Sullivan models in rational homotopy theory the crucial theorem which makes everything work is the one which says that the unit of the adjunction

Algebras $\to$ Spaces $\to$ Algebras

is, while not an isomorphism, a weak isomorphism, namely an isomorphism on cohomology!

So this means that when you start with a (nice) DGCA $A$ (a Sullivan algebra), then build a space $S(A)$ from that, and then take the differential forms on that space, $\Omega^\bullet(S(A))$, you’ll have a DGCA much larger than the original DGCA $A$.

But there is a canonical injection $$A \hookrightarrow \Omega^\bullet(S(A))$$ (which is the unit of the adjunction) and in rational homotopy theory one proves that this injection is a weak isomorphism an isomorphism in cohomology.

This is theorem 1.24 1), p. 10 in Kathryn Hess’s review.

(This works for Sullivan algebras, which are the cofibrant objects in DGCAs, as we discussed by email, last big paragraph on p. 6).

I ask because I couldn’t quite find this point explicitly dealt with in André Henriques’ paper either. Perhaps I have a misconception.

Last time I checked, which is a few months ago, it was not known to which degree precisely Ševera’s differentiation is inverse (left or right) to Getzler’s integration.

As I say, maybe that has changed by know. We’ll listen to what Pavol has to say in Lausanne!

Does this more pedestrian approach actually make sense, and can you show that this is the same thing (up to weak equivalence)?

I think heuristically this is precisely what the abstract nonsesense proceudure does for you.

Maybe have a look at section 5.2.1, p. 77, where I spell out how precisely the abstract nonsense of homming around reproduces exactly the constructions that Crainic and fernandes review in their review Integrability of Lie brackets.

Do these concepts like “tangent space” work out nicely for smooth spaces?

Well, the more immediate natural concept is cotangent space, which constitute indeed one part of the adjunctions, namely the object of infinitesima paths $$\Omega^\bullet : Spaces^{op} \to DGCAs \,.$$

Notice that the deRham DGCA $\Omega^\bullet(X)$ of a smooth space $X$ is the Chevalley-Eilenberg algebra of its tangent Lie algebroid $$\Omega^\bullet(X) = CE(T X) \,.$$