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January 28, 2008

Differential Forms and Smooth Spaces

Posted by Urs Schreiber

As we have discussed at length in the entry on Transgression, differential graded commutative algebras (DGCAs for short) have a useful relation to presheaves on open subsets of Euclidean spaces (smooth spaces, for short):

- every smooth space XX has a DGCA Ω (X)\Omega^\bullet(X) of differential forms on it;

- and every DGCA AA sits inside the algebra of differential forms of some smooth space X AX_A.

On top of that, every finite dimensional DGCA can be regarded as the Chevalley-Eilenberg algebra CE(g)CE(g) of some Lie \infty-algebroid gg, which linearizes some Lie \infty-groupoid.

Here I want to talk about my expectation that

The smooth space X CE(g)X_{CE(g)} associated to any Lie \infty-algebroid gg this way plays the role of the space K(G,n)K(G,n) # of the Lie nn-groupoid GG integrating gg.

As motivation and plausibility consideration, recall that in rational homotopy theory and notably in the work of Getzler and Henriques, one obtains a simplicial space S g S^\bullet_g from gg by defining its collection of nn-simplices to be the collection of gg-valued forms on the standard nn-simplex…

… and notice that with the above and using the Yoneda lemma, we can equivalently think of this as the collection of nn-simplices in X gX_g:

S g n=Hom smoothspaces(standardnsimplexin n,X CE(g)). S^n_g = \mathrm{Hom}_{smooth spaces}( standard n-simplex in \mathbb{R}^n , X_{\mathrm{CE}(g)} ).

Mapping simplices into a smooth space is like computing its fundamental \infty-groupoid Π (X CE(g))\Pi_\infty(X_{CE(g)}), thought of as a Kan-complex. In simple situations, notably when gg is an ordinary Lie algebra, also the ordinary fundamental (1-)groupoid Π 1(X CE(g))\Pi_1(X_{CE(g)}) is of interest. And I think Π 1(X CE(g))=BG \Pi_1(X_{CE}(g)) = \mathbf{B}G in this case, where the right hand side simply denotes the one-object groupoid with GG as its space of morphisms.

I am thinking, that hitting everything you see in sections 6 onwards in Lie \infty-connections (blog, pdf, arXiv) with gX gΠ (X g)g \mapsto X_g \mapsto \Pi_\infty(X_g) should have various nice consequences.

I want to better understand how nice exactly. That involves better understanding the properties of these functors DGCAs Ω () smoothspaces Π () inftygroupoids \array{ DGCAs &&\stackrel{\Omega^\bullet(--)}{\leftarrow} && smooth spaces \\ & \searrow && \swarrow_{\Pi_{\infty}(--)} \\ && infty-groupoids } in light of the above expectation.

All help is very much appreciated.

Posted at January 28, 2008 9:35 PM UTC

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13 Comments & 5 Trackbacks

Re: Differential Forms and Smooth Spaces

One thing I want to do — and I’d be glad to do it with you — is identify a class of ‘nice’ smooth spaces such that their deRham cohomology matches the real cohomology of their underlying topological space.

Then, given a ‘nice’ smooth group GG, I would hope to show BGB G is a ‘nice’ smooth space, whose cohomology is related to that of GG in the way we expect.

For example, starting from \mathbb{Z} we should be able to build up smooth spaces K(,n)K(\mathbb{Z},n) by iterating this BB construction.

I’ve got some of this working using Chen spaces. For example, I know how to form a Chen space BGB G from a Chen group GG. (I usually say ‘smooth’ instead of ‘Chen’, but I want to emphasize that there are various competing formalisms and I’m using this specific one.)

I haven’t made progress on isolating the ‘nice’ Chen spaces. But Mostow has an important paper on getting a smooth space BGB G for a smooth group GG, and working out its deRham cohomology. He uses a different formalism. Gajer has also done important work on making K(,n)K(\mathbb{Z},n)’s into diffeological spaces. So, I’m sure some idea like this will work. The challenge is to get a setup where it all works smoothly.

(Pardon the pun.)

Posted by: John Baez on January 29, 2008 6:00 PM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

Thanks for this reply.

I started writing a long reply, when I realized a stupidity I made: I kept saying that the presheaf X CE(g)X_{CE(g)} that I was talking about models BGB G. But if anything, it models K(G,1)K(G,1), right? And generally K(G,n)K(\mathbf{G},n) when gg is a Lie nn-algebra, following your explanation here.

Ah…

Posted by: Urs Schreiber on January 29, 2008 7:49 PM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

Following up on some hints dropped in Mostow’s paper, I’ve been looking at JW Smith’s work. It’s quite complicated, but it might be relevant for a couple of things on this blog.

The three papers that I think might be relevant are:

MR0202154 (34 #2027) Smith, J. Wolfgang. The de Rham theorem for general spaces.
Tôhoku Math. J. (2) 18 1966 115–137.

MR0145524 (26 #3055) Clifton, Yeaton H.; Smith, J. Wolfgang. Topological objects and sheaves.
Trans. Amer. Math. Soc. 105 1962 436–452.

MR0170314 (30 #552) Clifton, Yeaton H.; Smith, J. Wolfgang. The category of topological objects.
Proc. Nat. Acad. Sci. U.S.A. 47 1961 190–195.

For this thread, possibly the first is the most relevant. For the comparative smootheology discussion then the other two may be interesting.

I can’t say much more as I only have electronic access to the middle one and it’s quite complicated so I haven’t absorbed all of what the authors are doing. I think that my library has paper copies of the others so when I feel like braving the subzero temperatures and forging a path through the snow drifts then I’ll take a look at those and, if anyone else is interested, I could report back (but maybe someone else has electronic access to the other articles and will beat me to it).

Should I cross-post this in the smootheology thread? How does that work? Is that what ‘trackbacks’ are for? If so, how do I make one? ‘Scuse my divergent ignorance (which is even worse than unbounded ignorance).

Posted by: Andrew Stacey on January 30, 2008 9:02 AM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

Andrew,

thanks for all these references!

For this thread, possibly the first is the most relevant. For the comparative smootheology discussion then the other two may be interesting.

Strictly speaking, this thread here was supposed to be devoted to the approach involving presheaves “of plots” on abstract smooth test domains.

The articles you mention talk about sheaves of functions on a fixed topological space. Now, I am not really sure how directly this might be relevant for presheaves on abstract test domains.

Should I cross-post this in the smootheology thread? How does that work? Is that what ‘trackbacks’ are for? If so, how do I make one?

Trackbacks can only be issued by an entry, not by a comment, as far as I understand. I have in fact sent a trackback from here to Comparative Smootheology, but it might not have gone through yet.

I think maybe the best thing would be to simply post your comment separately again to Comparative Smootheology, if you feel it should be seen there, too.

even worse than unbounded ignorance

Hey, you were supposed to politely ignore that blunder, not post links to it! ;-)

Posted by: Urs Schreiber on January 30, 2008 10:47 AM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

I was particularly thinking of John’s comment:

One thing I want to do — and I’d be glad to do it with you — is identify a class of ‘nice’ smooth spaces such that their deRham cohomology matches the real cohomology of their underlying topological space.

This seems to tie in with the first of Smith’s papers - or so it seems from the review (haven’t been to the library yet).

Posted by: Andrew Stacey on January 30, 2008 11:42 AM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

This seems to tie in with the first of Smith’s papers

Yes, sure. All I meant was that Smith seems to be talking about sheaves of functions out of the thing, whereas my intention here (and I think John’s, too, since he was talking about Chen-smooth spaces) was to think about structures where we have maps into the thing.

But of course this shouldn’t stop us from thinking in all possible directions. In particular in light of the fact which you keep emphasizing, that we should be wanting to consider maps in and out anyway.

Posted by: Urs Schreiber on January 30, 2008 12:35 PM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

Urs wrote:

But of course this shouldn’t stop us from thinking in all possible directions.

I just want to say: YEAH!

It’s really too soon to say which approach to smooth spaces is ‘best’: the ‘maps in’ approach, the ‘maps out’ approach, or the ‘in-and-out’ approach, which is very popular here in Southern California:

Quite likely some combination of all 3 approaches is best. But, I think we need to set ourselves some nontrivial tasks, like getting a smooth BGB G from a smooth group GG and showing that its deRham cohomology works correctly… and a bunch of other tasks… before we have enough experience to make informed decisions.

Posted by: John Baez on January 30, 2008 6:35 PM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

set ourselves some nontrivial tasks

We have set ourselves the tasks. We need to make progress with solving them.

So let me proceed with the line of attack I was following:

I want to understand the set of closed differential nn-forms on the presheaf X CE(b n1u(1))X_{CE(b^{n-1}u(1))}. Due to the simplicity of b n1u(1)b^{n-1}u(1), this should be rather tractable:

a closed nn-form on this presheaf is on each test domain UU an assignment f Uf_U of closed nn-forms on UU to closed nn-forms on UU, natural in UU:

Ω closed n(U) f U Ω closed n(U) ϕ * ϕ * Ω closed n(V) f V Ω closed n(V) \array{ \Omega^n_{closed}(U) & \stackrel{f_U}{\to} & \Omega^n_{closed}(U) \\ \uparrow^{\phi^*} && \uparrow^{\phi^*} \\ \Omega^n_{closed}(V) & \stackrel{f_V}{\to} & \Omega^n_{closed}(V) } for each map ϕ:UV\phi : U \to V of test domains.

So in this simple case, a closed nn-form on the presheaf is precisely an endomorphism of the contravariant “deRham functor” Ω closed n:testdomainsSet \Omega^\n_{closed} : test domains \to Set which sends each test domain to the set of closed nn-forms on it.

I can currently see only the trivial endomorphism of this functor, which is possibly just because I am hoping not to see any further ones (always dangerous).

With a little luck you know some abstract nonsense that helps determine all endo natural transformations of this functor?

Posted by: Urs Schreiber on January 30, 2008 8:18 PM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

For example, starting from \mathbb{Z} we should be able to build up smooth spaces K(,n)K(\mathbb{Z},n) by iterating this BB construction.

To be more explicit, let me spell out my proposal for this situation:

Denote by b n1u(1) b^{n-1} u(1) the Lie nn-algebra of (n1)(n-1)-fold shifted u(1)u(1). The corresponding Chevalley-Eilenberg algebra CE(b n1u(1)) CE(b^{n-1} u(1)) is the differential graded commutative algebra which has just a single degree nn generator aa and trivial differential d CE(b n1u(1))a=0. d_{CE(b^{n-1}u(1))} a = 0 \,.

I think I am proposing that the generalized smooth space that plays the role of K(B n1U(1))=K(U(1),n) K(B^{n-1} U(1)) = K(U(1),n) is the presheaf X CE(b n1u(1)) X_{CE(b^{n-1}u(1))} which sends any test domain UU to the set of closed nn-forms on UU:

X CE(b n1u(1)):UHom DGCAs(CE(b n1u(1)),Ω (U))=Ω closed n(U). X_{CE(b^{n-1}u(1))} : U \mapsto Hom_{DGCAs}(CE(b^{n-1}u(1)), \Omega^\bullet(U)) = \Omega^n_{\mathrm{closed}}(U) \,.

The way to think of this is: there is a single closed nn-form aa on the space we are in the process of defining, and by choosing any plot from UU into that space we can pull this back to UU, and all closed nn-forms on UU can be obtained by such a pullback, uniquely. Hence a plot from UU into the space is the same thing as a closed nn-form on UU.

Accordingly, one finds that the space of differential forms on this presheaf Ω (X CE(b n1u(1))) \Omega^\bullet(X_{CE(b^{n-1}u(1))}) contains a closed nn-form.

I think actually that the cohomology of Ω (X CE(b n1u(1)))\Omega^\bullet(X_{CE(b^{n-1}u(1))}) is \mathbb{R} in degree nn and nothing else. But that I haven’t been able to show so far.

It’s the issue we were discussing before: for every Lie \infty-algebra gg there is canonically an inclusions

CE(g)Ω (X CE(g)) CE(g) \hookrightarrow \Omega^\bullet(X_{CE(g)})

which I would hope is actually an isomorphism on cohomologies. For the simple cases g=b n1u(1)g = b^{n-1}u(1) this shouldn’t be too hard to see, I am hoping…

Posted by: Urs Schreiber on January 30, 2008 10:33 AM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

Urs wrote:
“For example, starting from Z we should be able to build up smooth spaces K(Z,n) by iterating this B construction”.

To me, that looks more like applying your construction after iterating B.

Other than expressing things in terms of sheaves, why is it not enough to deal with simplicial manifolds to get smooth forms?

Then we could use the simplicial $B$ and iterate. Also one of the earliest realizations of $K(Z,n)$ was as a simplicial space.

As for: I think actually that the cohomology of Ω •(X CE(b n−1u(1))) is ℝ in degree n and nothing else. But that I haven’t been able to show so far.

Consider BBZ aka K(Z,2) realizable as the infinite complex projective space - I must be misunderstanding what you are trying to do.

Posted by: jim stasheff on January 30, 2008 1:24 PM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

Jim wrote:

Urs wrote:

For example, starting from Z we should be able to build up smooth spaces K(Z,n) by iterating this B construction.

To me, that looks more like applying your construction after iterating b.

Yes, in fact the sentence you quote was written by John #, and in my reply to it I didn’t exactly follow his suggestion, but tried to relate it to the construction that I keep talking about.

I must be misundertanding what you are trying to do.

Chances are that I am still confused about the right way to think of these presheaves. I was confused above and possibly still am.

So let me hand the question back to you all:

from any (finite dimensional) L L_\infty-algebra gg we can form a generalized smooth space X CE(g)X_{CE(g)} with the following properties (I think):

- The path groupoid Π (X CE(g))\Pi_\infty(X_{CE}(g)) has a single object and as such is the \infty-group GG that integrates gg.

- The Chevalley-Eilenberg algebra CE(g)CE(g) of gg sits inside the algebra of differential form on X CE(g)X_{CE(g)} CE(g)Ω (X CE(g)). CE(g) \hookrightarrow \Omega^\bullet(X_{CE(g)}) \,.

Question: which ordinary topological space does that mean is X CE(g)X_{CE(g)} somehow a “smooth model” of, if any at all?

Posted by: Urs Schreiber on January 30, 2008 3:03 PM | Permalink | Reply to this

Re: Differential Forms and Smooth Spaces

Recall that above and a few weeks back, I was looking for the right name for a space XX with the property that its fundamental \infty-groupoid Π (X)BG\Pi_\infty(X) \simeq \mathbf{B} G is a given nn-group GG. I oscillated a bit between thinking that X=K(G,1) X = K(G,1) or maybe X=K(G,n) X = K(G,n) and X=BG X = B G before becoming confused and handing that question back to the audience.

I must be being really dense here, but so be it.

Now I notice that Ronnie Brown in

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

does call such a space “B G”, in the proof of theorem 2.1 on p. 9.

Posted by: Urs Schreiber on March 5, 2008 11:06 PM | Permalink | Reply to this
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Re: Differential Forms and Smooth Spaces

The following just occured to me, which puts my statements here into a larger perspective by relating the smooth space which I am calling X CE(g)X_{CE(g)} not just to the classifying space of GG, but to the entire universal GG-bundle.

Recall that I am pointing out (more or less explicitly so far) that for generalized smooth spaces X, X \,, regarded as sheaves over smooth test domains, we can always form the (strict) fundamental nn-groupoid Π n(X) \Pi_n(X) in the essentially obvious way by looking at sheaf homomorphisms [0,1] nX [0,1]^n \to X and so on.

(In my latest article with Konrad we define in detail the groupoid P 1(X)P_1(X) of thin-homotopy classes of paths of any smooth space. This has a straightforward generalization to all other flavors of path (nn-)groupoids of XX.)

For my following comment I’ll just need Π 1(X)\Pi_1(X), the ordinary fundamental 1-groupoid of XX.

Now fix some Lie algebra gg and let GG be the simply connected Lie group integrating it.

Recall that I am writing X CE(g) X_{CE(g)} for the smooth space which is defined by the assignment UHom DGCAs(CE(g),Ω (X))=setofflatgvalued1formsonU U \mapsto Hom_{DGCAs}(CE(g),\Omega^\bullet(X)) = set of flat g-valued 1-forms on U and that I am claiming that the fundamntal 1-groupoid of this smooth space is

Π 1(X CE(g))=BG \Pi_1(X_{CE}(g)) = \mathbf{B} G

by which I use to denote the one-object groupoid given by the group GG. (This is supposed to be nothing by the de-nervified version of the standard construction described in On rational homotopy theory and Lie nn-tegration).

Now let

G set G_{set}

be the set underlying GG equipped with the discrete topology, regarded as the smooth space given by the assignment

U{constantGvaluedfunctionsonU}=G set. U \mapsto \{constant G-valued functions on U\} = G_{set} \,.

It is clear that

Π 1(G set)=Disc(G), \Pi_1(G_{set}) = Disc(G) \,,

where the right hand is supposed to denote the “discrete” category (in the cat sense, not in the topology sense) whose space of objects is GG and which has only identity morphisms.

Finally, write GG for the generalized smooth space given by GG itself.

Notice that

Π 1(G)=Codisc(G)=G//G=INN(G) \Pi_1(G) = Codisc(G) = G // G = INN(G)

is the pair groupoid of GG, since GG is simply connected.

Now, notice that we canonically have morphisms of smooth spaces

G setGX CE(g) G_{set} \to G \to X_{CE(g)}

The first map is just inclusion. The second map sends any function

g:UG g : U \to G

to its differential

g 1dg \mapsto g^{-1} d g

which is indeed a flat gg-valued 1-form.

Notice that all three smooth spaces we are talking about can be regarded as having functions that form a group. In that sense the above is actually a short exact sequence

G setGX CE(g). G_{set} \to G \to X_{CE(g)} \,.

By the above, hitting this with Π 1\Pi_1 yields the short exact sequence of groupoids

GINN(G)BG. G \to INN(G) \to \mathbf{B} G \,.

As discussed in some length at The inner automorphism 3-group of a strict 2-group

this is the groupoid version of the universal GG-bundle.

Posted by: Urs Schreiber on February 11, 2008 4:52 PM | Permalink | Reply to this
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