The n-Category Café

I propose that we use Richard’s explicit example he gave above as a good way for Urs, Jim and others to explain the bridge between the Chevalley-Eilenberg/$L_\infty$ approach and the Lie 2-algebra approach.

Let’s recall Richard’s question:

Here’s a very explicit example for everyone to sink their teeth into. Let’s think about a global quotient stack $[M/G]$. Here $G$ is a compact Lie group acting smoothly on a manifold $M$. The vector fields on this thing form a groupoid (page 33 of the same paper). Its objects are $G$-invariant vector fields on $M$. An arrow from vector field $X$ to vector field $Y$ is a smooth equivariant $\psi : M \rightarrow g$ such that $Y=X+ \alpha(\Psi)$. Here g is the Lie algebra of G and $\alpha : g \rightarrow TM$ is the map determined by the action.

Is this groupoid a Lie 2-algebra?

Let’s take it as reasonably clear that it does form a Lie 2-algebra. I am beginning to see/remind myself (since I once knew!) how it works in the language of Chevalley-Eilenberg algebras.

What is the 3-cocycle corresponding to this Lie 2-algebra? Is it nontrivial? Can it be written down explicitly in terms of the geometry of $M$ and $G$? Does the $L_\infty$/Chevally-Eilenberg approach considerably simplify this kind of computation, as opposed to the “pedestrian” approach of a Lie 2-algebra?

Hi John - at least in the algebro-geometric setting, these statements are all theorems. See in particular Theorem 3.2 of Hinich’s paper “dg coalgebras as formal stacks” which performs exactly this kind of integration in a different language. An even more general picture is explained on p.44 of Toen’s Higher and derived stacks, a global overview — at the time it was presented conjecturally but today I’m quite confident it is fairly straightforward.

In the differential topology context the language will be different but the fundamental concepts are the same: given an infinity-groupoid, or higher stack, one can differentiate it at a point to get an L_∞ algebra (or globally as an algebroid). Passing to the solutions of the Maurer-Cartan equation recovers the formal completion of the stack in question. This essentially follows from the Koszul duality between Lie and commutative operads and its ∞-version. The duality is given in one direction by passing to the tangent complex and in the other by formally exponentiating (aka passing to solutions of Maurer-Cartan equations, up to a shift).

How much of this is theorems?

That depends a bit on what you take take as definition.

David Ben-Zvi kindly recalls the algebro-geometric interpretation of this, which he first pointed out to me somewhere here on the blog back when I was running into this from my perspective in terms of “tangent categories” and “inner automorphism $(n+1)$-groups” and so on, as you will recall.

The theorems he mentions are based on some definitions people made and checked to coincide with the simple DGCA definition. But in the end one can turn things around and take the DGCA perspective as fundamental, I suppose. What one really wants is lots of consistency check that, whatever definition one settles on, it reproduces what it should reproduce in special cases.

And that it does. That what I call $inn(g)$ for $g$ an $L_\infty$-algebroid and what much of the rest of the world calls $T[1]g$ plays the role of the (shifted) tangent to $g$ (as the notation would suggest ;-) is taken as tautologous on all conferences with either “Poisson” or “BV” or “BRST” or the like in the title.

It reproduces the low dimensional examples one has and lifts them to familiar higher structures. The BV complex being the most striking example. There is a nice formulation of localizations of integrals on finite diemsnioonal manifolds obtained by passing to the shifted tangent bundle, and BV is just that generalized to $\infty$.

I am in a hurry now, so I will not give more details right this moment. But I’d be more than happy to discuss this stuff. In fact, I keep trying to do just that, for instance last time at Yet another model $\omega$-question.

before I have to run, let me just end by emphasizing, in case it is not clear, what some of the terms David B.-Z. used are in the language I have been using:

When David says “form the space of Maurer-Cartan elements” of $g$, then this is moving from the right to the left through the first step of this diagram, this space is $S(CE(g))$, the “smooth classifying space of flat $g$-valued differential forms”.

The “fomal exponentiation” then is forming the fundamental $n$-groupoid of this space, thereby moving all the way to the left, yielding $$\Pi_n(S(CE(g))) \,.$$ This is the $\infty$-groupoid $G$ integrating $g$. So one sees that $S(CE(g))$ is a smooth $K(G,1)$ or maybe better $K(G)$ space.

I am doing all this over test domains which are cartesian spaces, thus setting it up in the smooth context. To pass between that and David’s picture it should be true that we simply replace the site CartesianSpaces with $Rings^{op}$.

I think this is an important conversation to have. Urs has been trying to stress for a while now the whole $\infty$-groupoid, $L_\infty$-algebra, DGCA, BV, $\infty$-stack, $\omega$-groupoids side of things. (I write a whole bunch of buzzwords like that mainly because I haven’t grokked these things yet, so I just want to make sure I’ve covered all the bases! If I’ve left one out Urs, please tell me.)

I for one really need to get on board with this picture fast, because it is clear that this body of work has the potential of making so much of what some of us enthuse about at the n-category cafe seem so pedestrian and “old hat” :-) Not all of us mind you, because there are some around here who were involved in inventing this technology!

Imagine how silly we probably look to these people, getting excited about the fact that the tangent bundle of a stack forms a Lie 2-algebra, when it has already been done and dusted and well-understood for $\infty$-stacks!

I have tried to get on board with the ideas Urs has been pushing already, and I commit myself to learning this technology in the near future.

I think it’s very important that we understand just what kind of things are now well understood in that framework, and what kind of things aren’t.

David Ben-Zvi, would it be possible for you to give us your thoughts on what you feel are the outstanding issues at the moment in the game of “categorification”? I want to avoid working on something and having the nagging feeling that there is a whole bunch of people out there who would regard it as trifling. In fact that is already the case to some extent; when I look at Kevin Costello’s paper for instance my heart sinks when I think of what petty things I have been involved with :-)

John, I’d love to hear some wise words from you about this too. I think an important function of a blog like this is so that we can all get a feeling for what is known, what is not known, and most importantly of all, what people actually want to know.

Now I have the time to give a more detailed reply to the question

How much of this is theorems? #

It seems there is some room for interpretation what exactly “this” refers to, so I will say various things and ask you to get back to me with further questions in case I left out soemthing you wanted me to comment on.

So, I’ll talk about this diagram

and what I know about it for sure and what I expect to be true about it and what evidence I have.

I am setting this up in a smooth $\infty$-context using the model: $$Spaces = Sheaves(CartesianSpaces)$$ and $$\infty Groupoids = \omega Groupoids(Spaces) \,.$$ Notice that an $\omega$-groupoid internal to spaces is equivalently a sheaf with values in $\omega$-categories $$\omega Groupoids(Spaces) \simeq Sheaves(CartesianSpaces, \omega Groupoids(Sets)) \,,$$ which makes the contact to the stacky point of view that started this thread: a “smooth $\infty$-groupoid” in this context, i.e. an $\omega$-groupoid internal to $Spaces$ is a rectified $\infty$-prestack, if one insists on this point of view. This is sticking with our strategy to work internal to diffeological spaces, which, I think, is a good idea. Many things are treated as stacks for which this is overkill. I think $\infty$-Lie theory is such a case. But whoever reads this here and wants to have proper stacks instead of prestacks is free to apply stackification to the following statements.

Finally, $L_\infty$, the category of $L_\infty$-algebroids, is, by definition, the category dual to the category of qDGCAs (quasi-freee DGCAs): those coming from a chain complex $g$ of $C^\infty(X)$-modules as $A = ( \wedge_{C^\infty(X)}^\bullet g^*, d)$. We think of these as Chevalley-Eilenberg-algebras of the corresponding $L_\infty$-algebroids, which yields the contravariant functor $CE : L_\infty \to qDGCA$.

The above diagram arose as the result of my attempts to put into a single systematic picture the existing approaches to $\infty$-Lie theory which are out there. For the direction from right to left this is

a) Sullivan’s age old idea underlying rational homotopy theory

b) Getzler’s re-interpretation of that in terms of Lie theory (as summarized here).

Sullivan and Getzler take a DGCA $A$ and form the simplicial set/space $\langle A \rangle$ whose set/space of $n$-simplicies is the set/space of DGCA maps from $A$ to forms on the standard $n$-simplex:

$$\langle A \rangle_n := Hom_{DGCA}(A, \Omega^\bullet(\Delta^n)) \,.$$

All I observed is that it may be helpful understand this, trivially, as a result of a 2-step process:

first we form $S(A) \in Spaces$ given by $$S(A) : U \mapsto Hom_{DGCA}(A, \Omega^\bullet(U))$$ and then

second we take the fundamental $\infty$-groupoid of that space. For Getztler $\infty$-groupoid means Kan complex and fundamental $\infty$-groupoid means singular simplicial complex. Interpreting $\Pi_\infty(S(A))$ this way and using the Yoneda lemma produces the above expression $$\langle A\rangle = \Pi_\infty(S(A)) \,.$$

In my setup I use a different model for $\infty$-categories, namely $\omega$-categories, and hence a different model for the fundamental $\infty$-groupoid, namely $\Pi_\omega(X)$, the strict smoot fundamental $\infty$-groupoid whose $k$-morphisms are thin-homotopy classes of smooth images of the standard $k$-disk in $X$. So I form instead $$\Pi_\omega(S(A))$$ and its truncations $$\Pi_n(S(A))$$ obtained by taking equivalence classes of $n$-morphisms.

This is just a technicality, but I state it to make clear in what precisely the results are which I know. I think this is nice, because it produces results in a form which is close to stuff familiar in differential geometry and physics, largely because in the background we have Brown-Higgins theorem which says that $\Pi_\omega(X)$ is always equivalently encoded in a crossed complex of groups.

Okay, so here are the first results:

Let $g$ be an ordinary integrable Lie algebroid. Write $CE(g)$ for its Chevalley-Eilenberg DGCA (“the DGCA which computes Lie algebroid cohomology”). Then:

Theorem: $\Pi_1(S(CE(g)))$ is precisely the Lie groupoid integrating $g$ that is discussed in the literature, nicely summarized by Cranic and Fernandes.

In particular, if $g$ is an ordinary Lie algebra then $\Pi_1(S(CE(g))) = \mathbf{B}G$ is the one-object 1-groupoid coming from the simply connected Lie group $G$ integrating $G$.

(Here by “is precisely” I mean: the groupoid internal to sheaves which one obtaines is representable by the groupoid internal to manifolds that Crainic-Fernandes consider.)

This is proven by simply unwrapping (or maybe: “wrapping” ;-) what they are doing and checking that it amounts to precisely the computation of $\Pi_1(S(CE(g)))$, using the well known equivalences between Lie algebroid morphisms and the dual DGCA maps.

If the Lie algebroid $g$ is not integrable, then $\Pi_1(S(CE(g)))$ still exists as a Lie groupoid internal to Spaces, but now it is no longer living in $Manifolds \hookrightarrow Spaces$. We are precisely in the situation then that Chenchang Zhu considers in her articles #. I think it is clear that she essentially takes $\Pi_1(S(CE(g)))$ and stackifies its pre-stack of morphisms. I have not taken the time to write that statement up though, so you may want to feel cautious here.

Next there are the following integration results using this strict $\omega$-categoriecal integration

Theorem

1) For $g = b^{n-1} u(1)$ the $L_\infty$-algebra of $(n-1)$-fold shifted $U(1)$ we have $$\Pi_n(S(CE(b^{n-1} u(1)))) = \mathbf{B} \mathbf{B}^{n-1}\mathbb{R} \,.$$

(This is useful as an ingredient in many other computations. In particular, this facilitates understanding how $\infty$-Lie integration relates the model structures on DGCAs and $\omega Groupoids(Spaces)$ #)

For $g_{\mu_3}$ the weak String Lie 2-algebra with normalized canonical $3$-cocylce $\mu_3$, we have $$\Pi_2(S(CE(g_{\mu_3}))) \simeq \mathbf{B} String_{BCSS}(G) \simeq \mathbf{B} String_{Mick}(G) \,.$$ Here in the middle is our strict model of the String 2-group, on the right is a similar model using Mickelsson’s cocycle and the equivalences are 2-anaequivalences of strict 2-groupoids.

This is a joint result with Danny Stevenson. It shows among other things that the $\Pi_\omega$-integration procedure reproduces correctly the strictified versions of the known results, in particular with regard to André Henriques’ work. But moreover, this provides the starting point for eventually understanding Brylinki-McLaughlin’s construction of Čech cocycles as the computation of the obstruction of lifting $G$-bundles String-2-bundles. #

These examples (which in one way or other are known in the literature) serve the check that going from right to left through the above diagram is really $\infty$-Lie integration in that it reproduces in low dimensions what it ought to reproduce.

Next we want to understand the general $\infty$-Lie theory and then the differentiation procedure, to assign to each Lie $n$-groupoid its tangent Lie $n$-algebra.

So. One of the points of my 2-step reformulation is to make manifest that it consists of what should be two consecutive adjunctions.

If you look at the diagram above, you see that the step on the right is obtained by homming into/out of the “object of infinitesimal paths”, namely $\Omega^\bullet := U \mapsto \Omega^\bullet(U) = CE(T U)$, the tangent Lie algebroid.

Thanks to what Todd Trimble taught us, we know that the pair of maps obtained by such an “ambimorphic object” (deprecated original term: “schizophrenic”) constitutes an adjunction, being a generalization of Stone duality.

Now, I observe that the operation on the left also comes from homming into/out of an ambidestrous object, namely the “object $\Pi_\omega$ of finite” paths.

I think therefore that also the left pair of maps is an adjunction. This is kindergarten stuff for people like Todd.

Another thing which I have not shown but which I think is true, is that given these two adjunctions now, the process of going from left to right through this diagram is essentially what Ševera proposed in $L_\infty$-algebras as jets of simplicial manifolds.

Again, I can check the $\infty$-Lie differentiation process in low dimensions explicitly to see that it gives the right result

Theorem

1) For $X$ the discrete groupoid over space, we have $$\Omega^\bullet(K(X)) = \Omega^\bullet(X) = CE(T X) \,.$$

(So we do reproduce the Lie algebra of vector fields of ordinary spaces).

2) For $G$ a Lie group, we have $$\Omega^\bullet(K(G)) = \Omega^\bullet(S(CE(g))) \,.$$ (So we do reproduce the Lie algebra of a Lie group up to going once back and forth through the adjunction).

2) For $G$ a strict Lie 2-group coming from the differential crossed module $g$, we have $$\Omega^\bullet(K(G)) = \Omega^\bullet(S(CE(g)))$$ (so, again, we do reproduce 2-Lie differentiation up to going once back abd forth through the adjunction).

The proof of 1) is obvious, 2) and 3) are reformulations of theorems we proved in the context of parallel 2-transport #. These proofs follow an obvious pattern and it is clear how to continue them to higher dimensions. But this hasn’t been written up.

Also, the published proofs consider this just for Lie $n$-groups, not for Lie $n$-groupoids (though we do have material on those which didn’t make it our paper). But I think the proofs go through pretty much verbatim also for the groupoid case. This, too, needs more attention.

The fact that we don’t get the Lie $n$-algebra but something “larger” (forms on the classifying space) back from this general-nonsense way to $\infty$-Lie differentiation is a familiar phenomenon, already from rational homotopy theory, and appearing also in Ševera’s work. In rational homotopy theory the central theorem says that the canonical injection $$A \hookrightarrow \Omega^\bullet(S(A))$$ is a weak equivalence. I haven’t even tried to prove the analog of this for the context we are talking about (in rational homotopy theory this is a pretty nontrivial thing). But given that everything here is just a reformulation of basic ideas of rational homotopy theory, really, this is one statement which ought to be true. So this is a weak spot at the moment from the general point of view of $\infty$-Lie theory. But as far as I know, this is just as open in the formulation by Ševera (at least last time I checked, we’ll see how much progress there has been meanwhile).

Finally, I think I can prove the following, which is not tto hard:

Theorem: For $X$ a concrete space (i.e. a diffeological space) we have $K(\Pi_\omega(X)) = X$.

So on concrete spaces the left adjunction yields an isomorphism. (Notice however that all the spaces of the form $S(CE(g))$ are not concrete.)

Thanks for the detailed reply, Urs. I only partially understand it, but it’s a beautiful picture. I dream of having time to learn this stuff well enough to work on it. I’m not sure I ever will, and that makes me very sad, because this stuff — or something similar — clearly lies at the heart of 21st-century geometry.

Here’s a question or two.

You say you’re using ‘$\omega$-groupoids’. By this do you mean strict $\omega$-categories with strict inverses at all levels? One could also use strict $\omega$-categories with weak inverses.

My worry is that either way, these are ‘not sufficiently general’ — for example, ignoring ‘smooth’ aspects for now, not sufficiently general to model arbitrary homotopy types.

I know you’ve already pondered this. I guess you know that in 1991 Kapranov and Voevodsky once wrote a paper claiming that globular strict $\omega$-categories with weak inverses were sufficiently general to model arbitrary homotopy types. I guess you know that Carlos Simpson wrote a paper analyzing and in some sense disproving this claim.

Now, it’s easy to see why you’d prefer globular strict $\omega$-categories to Batanin’s globular weak $\omega$-categories, at least in the short term: even though they’re ‘insufficiently general’, they’re a lot simpler to understand and work with. So, they make a good testing ground for ideas that should be weakened later.

But if you want to think about $\infty$-groupoids, I don’t see why you’d prefer globular strict $\omega$-categories with strict or weak inverses to Kan complexes. The former are probably not sufficiently general; the latter clearly are sufficiently general. And, Kan complexes are probably easier to work with — at least after one learns all the techniques people have developed for them.

In short, I see Kan complexes and globular strict $\omega$-categories as good contexts for doing something less general than full-fledged $\infty$-category theory. But globular strict $\omega$-categories with weak or strict inverses seem strictly worse (pardon the pun) than Kan complexes.

Of course other considerations may push you towards them…