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Behind the scenes Domenico Fiorenza is having a long discussion with me and Jim Stasheff on the matters that are being discussed at differential cohomology in an (oo,1)-topos – examples. It seems we want to work on this together. Accordingly, I have now moved at least parts of this to the main nLab in the new entry
I added a remark right at the beginning that is supposed to indicate the nature of this material.
From the entry it is not clear even a hint of what are the remaning (edit: targetted) open problems in this program.
Good point, I’ll add a section on that right now.
okay, I started writing a section Motivation.
as usual, there is plenty of room for improvement, but I need a short break now…
the discussion at infinity-Chern-Weil theory builds on the discussion at circle n-bundle with connection. So I thought I might move that over to the nLab, too. And I did. There is a paragraph at the beginning indicating the nature of this material.
(This is what I talked about in Copenhagen two weeks ago, by the way).
To begin with I’ll be extremely annoying.. by reading line by line the Motivation section, and asking for expalnations everytime I don’t understand something.. :)
At the beginning of the more technical part, it is said that ordinary Chern-Weil theory provides a refinement of the fractional Pontryagin class
to a map to ordinary differential cohomology
what is exactly meant by this? i.e., what is it meant by ? I followed the link to differential cohomology but got lost there, without being able to decide what stands for :(
A few lines later and seem to be to two symbols for the same object.
Let us be down to Earth. The usual Chern-Weil theory produces the characteristic classes for principal and associated G-bundles where G is a subgroup of GL(n). The classes are in even dimension. Chern-Simons and similar classes are secondary characteristic classes, and they come in odd dimension, and they are not produced by the classical Chern-Weil theory; on the other hand they are related to the differential characters of Cheeger and Simons. Now, my understanding is that going to infinity Chern-Weil theory is to have the usual characteristic classes like Chern classes, Todd class, Stiefel-Whitney class on the same footing with the secondary classes and go beyond. Most of classically educated geometers will start from the position I am having in this question. So where is seen that going to (infinity,1)-world one goes beyond the classification of (primary) characteristic classes and has more. I understand and remember all the examples string fivebrane and all that, but I would like to see them as some sort of conceptually simple extension of classical framework which avoids the theorem that the only characteristic classes are power series in known classical classes, and in particular only even cohomology is involved.
what is meant by .
The reply to this question is now at ordinary differential cohomology.
The usual Chern-Weil theory produces the characteristic classes for principal and associated G-bundles
Let’s be more precise: it produces refinements of characteristic classes.
For , and a -principal bundle on , the bare characteristic class is .
What the Chern-Weil homomorphism does is refine this integral class by providing a differential form that represents its image in real cohomology.
And more is true: if the class on refined to a class in differential cohomology then the Chern-Weil homomorphism may be used to produce accordingly a class
that refines this integral class.
A few lines later and seem to be to two symbols for the same object.
Sorry, I don’t see which sentence this is referrin to right now. Likely I made a typo somewhere, but could you check again. What’s the beginning of the sentence that you mean?
Chern-Simons and similar classes are secondary characteristic classes, and they come in odd dimension, and they are not produced by the classical Chern-Weil theory;
They are produced by the refinement of Chern-Weil homomorphism to differential cohomology. This is maybe not very classical, but still pretty classical (for instance constructed explicitly by Brylinski-MacLaughlin):
for a given invariant polynomial, the Chern-Weil homomorphism may be thought of as producing the -gerbe with connection whose curvature is the corresponding curvature characteristic form, and whose connection is the corresponding Chern-Simons form.
And the buisness about “secondary” characteristic classes just says that if the curvature happens to vanish, then already the connection form is closed.
So where is seen that going to (infinity,1)-world one goes beyond the classification of (primary) characteristic classes and has more.
I tried to say this clearly in the Motivation section, but if I failed this needs to be stressed more:
ordinary Chern-Weil theory deals with characteristic classes of ordinary G-principal bundles, i.e. of the classifying spaces for an ordinary Lie group. Higher Chern-Weil theory generalizes this to characteristic classes of -principal -bundles for an -Lie group.
I said:
ordinary Chern-Weil theory deals with characteristic classes of ordinary G-principal bundles, i.e. of the classifying spaces BG for G an ordinary Lie group. Higher Chern-Weil theory generalizes this to characteristic classes of G-principal ∞-bundles for G an ∞-Lie group.
Okay, I added this kind of statement more explicitly at the very beginning of infinity-Chern-Weil theory.
@Urs, #8:
Clear. I have to ask another thing on this, but what I will ask depends on your reply on #10 :)
@Urs, #10:
the sentence begins with: The first point of passing to a higher category theory-refinement of this situation is that it allows to refine, in turn, this morphism of cohomology sets to a morphism
Thanks, I understand that infinity theory in general applies to infinity bundles but the 2-group responsible for gerbes for Chern-Simons is coming from a suspension of 1-group, so the classical point of view is still from 1-group, thus “secondary classes”. When I told one excellent mathematician about your work few months ago, mentioning the refinements of classes classifying higher bundles, then his first question was if you have (the generalized) secondary classes for those. So you see, the impression from the classical geometer is that one looks all from the point of view of 1-groups and 1-bundles, even if the more natural picture is as you said in different terms. So I wanted and I am partly but not fully answered how to see the need for and the repair by the infinity theory to the classical problems. ( In classical problems even the bundle has been usually understood (just the tangent bundle) so one talks of functorial assignment of a cohomology class of a manifold functorially to a manifold. )
From the classical point of view, one looks at the characteristic classes in ordinary cohomology and after the Chern-Weil procedure the obtained class does not depend on the choice of connection. This independent result is what classically one called the Chern-Weil construction. Now in the big of the things you are doing all these are special cases, so you renamed the procedure a bit and call the Chern-Weil procedure the maximum you can get from the algorithm (what seems to neglect in part the remarkable fact that the usual class does not depend on the connection).
One unrelated thing bothers me: for vector bundles one always have subgroup of unitary or orthogonal group. But for principal bundles one can look at bundles whose structure group is a non-linear Lie group (hence not a subgroup of GL(n)). My vague memory is that many classical theorems about using Weil algebra and invariant polynomials depend on the structure of linear groups. Is there a problem or I am just halucinating ?
Domenico,
conerning that typo: yes, that was indeed a typo. Thanks. I have fixed it and added a bit more of clarification (hopefully).
Zoran,
his first question was if you have (the generalized) secondary classes for those.
I haven’t used much the term “secondary class” (maybe I should) but I think this is all contained in the consztruction.
In fact, those “differential string structures” for instance are really refinements of the “secondary characteristic class” corresponding to the first Pontryagin class. Because “secondary class” means: the curvature class itself vanishes.
Okay, so I take the point that I should discuss the relation to the notion of “secondary classes” more explicitly. Thanks.
how to see the need for and the repair by the infinity theory to the classical problems.
But what about this: classical Chern-Weil theory only exists for characteristic classes defined on classifying spaces for a Lie group. Isn’t it in itself a motivation to generalize this to more general spaces? That’s what the -theory does.
And then there are examples of problems where one does want to have a Chern-Weil theory for characteristic classes on a space for not a Lie group, coming from physics. So it seems to me we have both a general mathematical as well as a more contrete motivation from explicit applications.
This has been, after all, the driving motivation all along for the whole effort, starting with Stolz-Teichner’s question: What is a connection on a String-bundle?
( In classical problems even the bundle has been usually understood (just the tangent bundle) so one talks of functorial assignment of a cohomology class of a manifold functorially to a manifold. )
Right, that’s an point that I should discuss more: what effectively happens there is that one talks about the classes of the -bundle, even if that lifts to a -bundle for somewhere higher in the Whitehead tower.
While one can do this, this misses information: for instance on we have the ordinary first pontryagin class , but up on this can be divided to a “smaller generator” in that
And this continues: next one has that the second Pontrygin class may be cut down by 6 even when passing to String
So if we are on a String-manifold and treat it just as an ordinary oriented manifold using its tangent bundle, then the second Pontryagin class that we compute is six times “too coarse”: for instance if the 8th integral cohomoly of the manifold is 6-torsion, then that naive class will vanish. But the refined class produced from the oo-Chern-Weil homomorphism will not!
But you are right, this kind of dicussion should be added to the motivation section.
Fine, but shouldn’t we have some differential refinement on the left-hand side, too? (I can imagine a differential nonabelian describing isomorphism classes of principal bundles with connection). Otherwise I don’t understand how the refinement (i.e., the connections) arise on the right-hand sides. For instance, what does happen if one starts from the beginning with and with the fisrt Chern class and goes on along the lines sketched in the Motivation section?
Fine, but shouldn’t we have some differential refinement on the left-hand side, too?
Right, so that’s an important point, already in the ordinary Chern-Weil theory, but same here. The story is this:
As you suggest, let me first recall how is actually obtained:
for that abelian case, there is a canonical and essentially unique curvature class,
This exists on general abstract grounds. If you model this morphism in some 1-category, then you find that this morphism operates by choosing a connection on a circle -bundle and then pcomuting its curvature. The differential cohomology is the homotopy pullback
This construction is not available for general coefficients . Because in the bottom right we would have , and that double delooping for general does not exist.
So this is why “nonabelian differential cohomology” is a bit different from “abelian differential cohomology”. The latter has an exact obstruction theory to flat connections, the former not.
But what we do have is detailed approximations to the non-existing exact curvature class. Namely we can take the nonabelian and sort of approximate it by lots of abelian things.
More precisely: we can take any abelian cocycle
This sends -principal bundles to -principal bundles. Then we can take the curvature classes of these . That’s just postcomposition with the total composite
.
So we have morphisms
and
By the -pullback definition of we have that these two morphism together induce a morphism
Now again, when we model this abstractly defined -functor by 1-categorical models, we find that a miracle happens: this functor implcitly operates by first choosing an -connection on a -principal -bundle, then computes its curvature forms, and then evaluates these in a an invariant polynomial to produce a single -form.
So as before in the definition of , the connection is something that appears as an intermediate step in computing the curvature classes.
That means we find the connections in the homotopy fibers of these morphisms. It is the cocycles in the homotopy fiber of that encode a differential refinement of .
Very sensible answer, thanks Urs. (Edit: Urs posted another answer in the meantime, to Domenico’s question. I was about the previous one.)
I have to think more on this.. what confuses me is the basic example where one seems to end up with a morphism which I’m unable to see as something canonical. I see I have to study curvature classes of -bundles..
How is the morphism defined? Such a morphism would induce , and so , i.e., since it seems can be an arbitrary object in , a morphism , which I don’t clearly see (I can imagine various “natural” such morphisms, but no canonical one). I’m very confused, so this question is probably totally nonsensical: don’t pay too much attention to it.. better I go to sleep and think back to this tomorrow.
In reaction to Zoran’s remarks above, I have now expanded the Motivation-section further. It now has two subsections, of which the first is the new one. This discusses a bit the issue of fractional characteristic classes and how oo-Chern-Weil theory sees them, while ordinary Chern-Weil does in general not.
what confuses me is the basic example where one seems to end up with a morphism which I’m unable to see as something canonical.
Yeah, it takes a bit getting a feeling for these -functors.
The intuitive way to think about is this: there is a universal connection on the universal -bundle . (In classical literature this is constructed as an ordinary Ehresmann connection on an infinite-dimensional manifold model for B U(1) .) This morphism takes a smooth cocycle for a smooth -principal bundle and pulls back the universal connection along it.
A concrete way to think about it is this:
it is a theorem (proven at circle n-bundle with connection) that this -functor is modeled in the model structure on simplicial presheaves by the span (anafunctor)
where the right morphisms is a fibration, and where is a the simplicial sheaf
( the Dold-Kan map that sends sheaves of chain complexes to sheaves of Kan complexes).
When one thinks about it (detailed discussion at the above link) one finds that this is the sheaf whose hypercohomology are -principal bundles with “pseudo-connection”.
The curvature of a pseudo-connection need not be a globally defined 2-form, but is a general cocycle in the complex .
But in any case, the important point is that taking morphisms into account, -principal bundles with pseudo-connection are equivalent to -principal bundles without any connection! That’s just restating that the left leg of the above span is a weak equivalence.
So for computing the defining -pullback
we may just compute the ordinary pullback
where the angular brackets denotes the hom-complex of simplicial presheaves, and where denotes a cofibrant resolution of , namely the Cech nerve of a good open cover.
Here the right vertical morphism picks one point in each connected component. Since the oo-groupoid on the right has connected components equivalent to the second deRham cohomology of , we may without restriction assume that the right morphism picks cocycles that happens to be given by globally defined closed 2-forms.
With that setup (which we know is one way to compute the abstractly defined -pullback in question) we find that the pullback picks among all pseudo-connections (this are the cocylces in the bottom left) precisely all the ordinary connections.
This way, we find that is line bundles with ordinary connection.
Don’t know if it helps you, but I found it useful to meditate about this explicit computation of the abstract oo-pullback that defines the intrinsic differential cohomology a bit. I think this (simplest nontrivial example!) sheds a bit of light on what differential cohomology is all about. See circle n-bundle with connection for a bit of exposition and some comments on this.
(Now the above comment seems to display all diagrams. I had to change all double dollar signs to single ones. )
How is the morphism defined?
Probably you mean the morphism .
This is defined by the following pasting diagram of -pullbacks
Here and is the counit of the -adjunction . The object is by definition the homotopy fiber of this morphism. The remaing diagram then follows from the pasting law for -pullbacks.
Concrete examples for are worked out at infinity-Lie groupoid. For an ordinary Lie group, this is the Maurer-Cartan form. For this morphism is given by the span that first puts a pseudo-connection on a circle n-bundle and then project out its curvature cocycle.
How is the morphism defined?
Never mind: as I was suspecting in the last part of my question, I was just halucinating this morphism :) I was reading the section on intrinsic de Rham cohomology in your area and somehow at a certain point I read in place of , and in such a sleepy state as I was, instead of just having a second look I started speculating… this cannot be, the natural arrow should be the other way round..etc. As I wrote my question, mostly to clear my mind, I looked back and found my mistake, but mysteriously in that precise moment nForum decided not to allow me edit my question or write anything anymore :(
Luckily, you were able to make something good of it anyway: thanks for the detailed answer on the morphism ! :)
@Urs, #24
Don’t know if it helps you
It helps a lot, indeed! I think I’ll devote the whole day to the deep understanding this basic example.
The intuitive way to think about is this: there is a universal connection on the universal -bundle.
Let’s remain at an intuitive level to see how much I can get of the general feeling from this. Clearly, being a universal object in the category of -bundles with connections is a universal property, so it is perfect to say the universal -bundle with connection. On the other hand, if we fix a model for the universal -bundle and then we endow it with a connection, we still obtain a model for the universal -bundle with connection, but a choice has been made. Once this choice has been made, every morphism can be promoted to a -bundle with a connection over . If we change our choice, then we will end up with “different but equivalent” connections on our -bundles: in other words, if we work at the level of universal construction, then everything is canonical, when we pick up concrete models, then choices come in sight and the canonicity of the construction is obscured.
mmm.. I see.. my confusion comes from just an habit of thinking of many different concrete models here rather than in terms of universal properties. since these different concrete models are all on an equal footing not only from a mathematical perspective but also from an educational one, this in confusing. but actually the same phenomenon should happen for every universal property: for instance when considering the product of two sets and , I’m so trained to consider the set of pairs that I would not question that this is not canonical but a choice! for connection I have not been educated to an historically preferred model, and that’s it! ok, now I think that going back to what you wrote and studying it I can undersatand it :)
Its certainly important to amplify the role of models in this business, as compared to things with intrinsic meaning.
Already for the universal bundle, this is an issue that deserves highlighting: in intrinsic terms, the universal bundle is an empty tautology, it is just the point. It gains its meaning only after we fix a model and work with specific choices.
I think for universal connections its the same kind of situation. Intrinsically the concept either does not exist or is tautological and empty, depending on your point of view. But it becomes important when concrete models are chosen. But then it does depend on lots of choices, and only when we go back to the larger intrinsic perspective do we see that all is fine and well.
At infinity-Lie groupoid I have a section with a discusson of the universal conneciton on the universal G-bundle in a manner somewhere half-way between the complete abstract nonsense and the models by manifolds in the literature. The model discussed there has a certain appeal – to me – in that it identifies the universal connection on a groupal model (groupal model for universal principal infinity bundles) for as the 2-Maurer-Cartan form on this 2-group.
I think this is useful for understanding -Ehresmann connections, but for the moment the discussion of that is a bit incomplete.
In any case, all that is not necessary for our discussion here, but might provide some useful extra perspective.
it is just the point
Well, universal element. So if the representation (the structure of representable functor with the choices fixed) is fixed, it is unique.
I am not quite getting the rest, are you claiming that the universal connection is also part of the data of a universal element for refined moduli problem ?
Minimal language suggestions inthe section “Higher differential string structures”: maybe we could say “Then ordinary Chern-Weil theory provides a lifting of the fractional Pontryagin class” with lifting instead of refinement, and replace the two one-horizontal-arrow oo-categorical refinements (I would leave refinement here) just below with a single triangular diagram exhibiting as a lift of .
A little later in the same section is used. By analogy with the diagram above the one where the symbol is introduced, one has , so would seem a natural choiche to denote the pullback denoted . I can imagine the symbol is not used since the refinement is not available for nonabelian groups, but I find the use of so diffeent notations in the two situations making things more obscure than necessary.
I’m working out my homework.. is a 3-groupoid or a 4-groupoid? at infinity-Chern-Weil theory, in the Higher differential string structures section we’re saying 4-groupoid, but I would say 3-groupoid because going down from 3 to 1 I would expect that is a 1-groupoid rather than a 2-groupoid. On the other hand, I see would appear because of , and so a 2-groupoidal interpretation would be fine. How should I think to this?
Well, universal element. So if the representation (the structure of representable functor with the choices fixed) is fixed, it is unique.
What I am saying is that homotopy-theoretically, every principal bundle is the pullback of the point . The homotopy pullback. So that universal element is rather trivial.
with a single triangular diagram
Which triangular diagram exactly do you envision here?
I can imagine the symbol is not used since the refinement is not available for nonabelian groups,
Yes. Or maybe more precisely: it depends on which curvature classes we use. The objects of the homotopy fiber of may be identified with spin-bundles with connection, but the morphisms may contain shifts of these connections that only respect some condition on the 4-form curvature and nothing else.
a 3-groupoid or a 4-groupoid?
You are right, it is a 3-groupoid.
Concerning versus : in the oo-topos of bare oo-groupoids these are equivalent, but their lifts and to the oo-topos of oo-Lie groupoids are not equivalent there. So the difference matters.
For talking about differential refinement that way we are, it is crucial to use .
this triangular diagram:
where the vertical arrow forgets the connection.
the corresponding diagram for in place of still causes a few problems to me: I’m unable to clearly see a section to . Or better, I do see a section if I consider and as abstract groupoids, without trying to give a geometrical meaning to this section: just pick a representative for each isomorphism class in , choose a connection on that and then “propagate” this connection along isomorphisms of -bundles to get a functor . Is this correct?
concerning the diagram: okay, feel free to insert that.
concerning those sections: you are right, I think I made a wrong statement here. What we get from morphisms is morphisms
and
(where non-boldface means connected components, as usual)
but not in general
Sorry, somehow my mind played a trick on me here. But wait, let me think.
Domenico,
good that we are talking about this. Let me try to fix this.
So back to that universal connection: let me argue for a morphism
This is supposed to be induced from the diagram
where the left 2-cell is the identity and the thing is that the triangular 2-cell in the top right should exists, because has no real cohomology below degree . (The failure of this statement for general is what makes the corresponding argument in this fashion for general fail ).
So then the commutativity (up to homotopy) of this digram produces, by the -pullback-definition of a morphism
This in turn gives a map from morphisms to morphisms
as morphisms of -pullbacks induced by th morphisms of diagrams that is induced by precomposition with .
This should be functorial in . Forming the image of under the composite
should then give the dersired morphism.
What I am saying is that homotopy-theoretically, every principal bundle is the pullback of the point . The homotopy pullback. So that universal element is rather trivial.
Its total space is pullback of the point. But as a bundle with all the structure it is of course not trivial. So I am not fully getting the claim that it is interesting only in a specific model, as there are possibly nonisomorphic bundles on BG in any setup and EG -> BG belongs to a rather special class.
EG -> BG belongs to a rather special class.
That’s true, but that’s because it is a nice model. What matters is that is a fibration that resolves . That alone is sufficient to guarantee that the ordinary pullback of is the homotopy pullback of the point. And from the fact alone that something is the homotopy pullback of follows that it is a principal oo-bundle.
If you start with an ordinary group G and start with a fibration replacement that is not a universal bundle in the usual sense, the resulting pullbacks will still have a principal G-action up to coherent homotopy.
A striking example of this is the theory of universal simplicial principal bundles (as summarized there):
we know that simplicial groups happen to be a semi-strict model for general oo-groups. A simplicial principal bundle is accordingly a very strict model for a principal oo-bundle. It turns out that the usual theory of universal 1-bundles goesthrough essentially degreewise to produce a theory of universal simplicial principal bundles with strict degreewise group action. But of course that’s manifestly just a property of a nice model: nothing in the homotopy theory of oo-groups requires that we use such strict models. The general principal oo-bundle of course can have and will have very non-strict action.
Domenico,
I have thought a bit more about it and think my above answer is correct. In particular I think I have a proof that indeed we have these 2-cells
expressing the fact that the real cohomology of is concentrated in degree .
So the -categorical construction of
goes through. I think. What I need to think about is what you kept asking me: what this general abstract morphism looks like in our concrete favorite model, in particular of course what it does on higher cells. I had underappreciated this before, having concrentated on just the curvatures.
I will think about this and get back to you. But now I have to sleep and tomorrow morning I have to meet a student. Then I need to hop on a train. So it might not be until tomorrow evening or so that I am able to get back to you with substantial remarks.
Hi Urs,
sorry for replying so late: I’ve been out this evening, just come back. Tomorrow evening is just perfect: I’ll work on yours #37 and #41 during the morning, and I’ll be ready for the evening :)
How is the morphism defined?
what I still don’t clearly see in the differential cohomology refinement is the fact one makes all the construction be carried by the abelian side of the picture. At least in the usual context of principal bundles with connections (with a Lie group), I would think of a differential refinement as the groupoid of pincipal -bundles with a connection. Then we would have a natural forgetful map . How is this related to the abelian business? let me sketch what I have in mind for , so that whe meet a characteristic class already in and need not go into the realm of higher groupoids (not that this would make any actual difference, but just to keep the example at the lowest level). Let be the first Chern class. Then we have for any nice topological space the first Chern class in cohomology: . If is a differential manifold, then is the set of isomorphism classes of smooth -bundles on , and is the set of isomorphism classes of smooth -bundles on . Moving from isomorphism classes to groupoids, we can therefore think of the first Chern class as a map
The first question arising here is whether this map at the level is induced by a morphism of groupoids
the answer is yes! (and this morphism is nothing but the delooping determinant map ). Now classical Chern-Weil(-Deligne?) theory comes in: let be the groupoid of principal -bundles with connection on . Then we have forgetful maps
and
and one may wonder whether lifts to a map
making the diagram commute (up to a given natural transformation, of course). Here I don’t know the answer, but I guess it is yes, and that (the classical version of) nonabelian differential cohomology is all about this (in this totally basic example, I mean). Postcomposing with one gets the classical Chern-Weil construction mapping a -bundle with connection to its curvature 2-form.
Now, what I think is a crucial question here is whether the diagram
is a pull-back diagram. If yes, then we could just define as the (homotopy) pull-back of
thus defining a notion of principal -bundle with connection suitable for immediate generalization to higher groups: only thing one needs is to fix a cocycle to define the lower horizontal map in the pull-back diagram).
How is the morphism defined?
By choosing one element in each connected component.
Here is the idea: a homotopy fiber of a morphism is defined with respect to a point in . But up to equivalence, the homotopy fiber depends only on the connected component that hits. For differential cohomology we want to find all the homotopy fibers of . So we want to pull back all possible maps . Since only the connected components these hit matters, we choose one for each connected component.
This point uses to lead to some discussion. People usually ask me why I don’t consider the map instead, that includes the set of objects. This is in fact what happens in the ordinary definition of differential cohomology: in standard situations we may identify this set of objects with the set of closed forms, and inject these.
We can do this, indeed, once we fix a model. But the result will depend on that model, since the “set of objects” of an oo-groupoid is not a notion defined up to equivalence. It is evil. But also the difference between the two versions is not essential, since, up to equivalence, the homotoppy pullback of any one object depends only on its connected component.
what I still don’t clearly see in the differential cohomology refinement is the fact one makes all the construction be carried by the abelian side of the picture.
For each notion of “curvature characteristics”
we get a notion of differential refinement for it: its homotopy fibers.
If is deloopable, there turns out to be a “good” such curvature class, in the sense that i computes precisely the obstructsions to flatness. So that’s an important special case.
But not the only case. We can consider other maps, but they are more involved. In the fully general case, we can consider the nonabelian Chern character
which is the composite of the canonical inclusion where is the geometric realization of , with the rationalization map (over the reals) .
This induces a map
and I think the full notion of nonabelian differential cohomology is the homotopy fibers of that.
But this general map is hard to get under control. We can try to conquer it by breaking it up into pieces, though, using Postnikov-Whitehead decomposition
Let
be the Postnikov tower of . Then by iteratively forming pullback, we get a tower of characteristic maps out of the Whitehead tower of
along the lines we once discussed at Whitehead tower in an (infinity,1)-topos.
So this gives a tower of abelian curvature classes, which we can each handle by the abelian theory. I think under mild conditions knowing all these abelian differential cohomology classes on all these Whitehead items allows to reconstruct the fuill nonabelian Chern-character map.
For instance for we see the tower of Pontryagin classes appear this way. The following diagram gives an impression of a smal part, just because I happen to have the source code for it lying around for quick copy-and-paste
Similarly for one would get the tower of Chern classes this way. I think one needs all of them to characterize a sensible nonabelian differential cohomology.
I think your suggestion of pulling back just the first Chern class will give a notion of connections on -bundles where two are identified if shifting from one to the other leads to a morphism of the correspond for Chern class line bundle with connection.
I think an important concept to look at in this context is Simons-Sullivan structured bundles. If you look at that, you see that they effectively look at connections on -bundles modulo the relation that identifies two if the shift between them induces a morphism of all the corresponding “lifting”-circle n-bundles with connection.
In the entry of oo-Lie groupoids I mention a way to reproduce this in the present context: by decategorifying the full Chern-character map a bit, one can get a morphism
to a smooth groupoid whose objects are smooth closed differential forms, one for each invariant polynomial, and whose morphisms are degree-minus-one forms modulo exact forms. The map sends a (pseudo-)connection on a G-bundle to the collection of its curvature invariant forms, and a transformation between these to the corresponding Chern-Simons forms . These are indeed well defined modulo exact terms, so this does give functor.
Then the ordinary pullback of points along
does reproduce the groupoid of Simons-Sullivan structured bundles: its objects are (ordinary) connections on -principal bundles, but morphisms are allowed to be morphisms of bundles that shift the connections, but only so that the corresponding Chern-Simons forms of the shifts are exact.
My impression is that this kind of construction is what you seem to be after. But I also think that this is still just an approximation to something, becuase it involves those decategorifications etc.
For differential cohomology we want to find all the homotopy fibers of..
I can remember us discussing this at twisted cohomology. There we arrived to something definite which is precisely what you recall to my memory now: one chooses a representative element for each to define -twisted cohomolgy. I can remember this choice issue not completely satisfying me, but I was unable to see anything better at that time, and ended up with convincing me that was optimal. But now that I look back at it, I think: the point in twisted cohomology is to consider all possible homotopy fibres of . And homotopy fibres are nothing but (or better, can be seen as) fibres of a fibrant replacement. So why don’t think to the whole of all twisted cohomologies just as a fibrant replacement of ? This way one should get a morphism which is a fibration, and would be the twisted cohomology associated with .
For instance, one would have as the fibrant replacement of .
Similarly for one would get the tower of Chern classes this way. I think one needs all of them to characterize a sensible nonabelian differential cohomology.
Sure: my suggestion was not to use only : what I wanted to show in that oversimplified example was how there was a quite natural point of view not involving choices of representatives for each connected component.
Not a reply to what you just said but an addendum to my previous message, before I have to rush off:
from a topos-theoretic perspective, the following looks natural: we are looking for a canonical morphism that takes -cocycles to differential forms with values in something.
Now is a right adjoint, its left adjoint being . This means that we always do have a canonical morphism
I used to think that this is the curvature class that one should consider for nonabelian differential cohomology. I spent a huge amount of thought into understanding this, but at some point I decided that I was not succeeding and should look at something simpler for the time being.
But eventually I would like to understand the homotopy fibers of the curvature classes induced by this canonical map. Abstract nonsense tells us that this is a good thing to look at.
What is ? we have a definition at Higher differential string structures, which presently depends on the map . this has been a little trouble since according to post #36 there should be no such map, but then according to post #41 this map should actually be defined. what i do not see yet is the datum of a connection on the Spin bundle which one finds in the explicit description of one finds just after its homotopy pullback definition. So I guess that the datum of is hidden in the fact one has promoted to . however I’m unable to convince myself this is meaningful: how can changing a morphism out of a fixed category change the objects of that category? I mean, if objects in where -bundles without any connection, I still don’t see how the datum of a connection can appear when one changes the target of a morphism out of . What continues to seem a natural solution to my doubts is to consider the groupoid of Spin-bundles with connection in place of . A few posts ago I suggested to call this , but I strongly suspect this would not be a fine name, since refers to a general construction which does not applies to nonabelian groups. But then maybe could be a plausible name.
I spent a huge amount of thought into understanding this, but at some point I decided that I was not succeeding
What aspect you are not happy with here ?
Hi Domenico,
as promised, I am in the process of spelling this out. As announced, I was busy part of the day. But now I am getting back to you.
For the moment, I have completed the first step and spelled out the proof that
.
This is now at oo-Lie groupoid – de Rham cohomology of Bn U(1). This implies that the general abstract construction of the morphism works. Next I need to spell out what it is concretely.
While I am doing that, to answer your question how it can be that objects in the homotopy fiber of a morphism out of to anywhere can know about connections on -bundles: let’s for the moment look at this in the cases where the full proof is spelled out (while I am fixing the gap in the one in question).
Do you see how it works for the case of spelled out at circle n-bundle with connection?
The thing is that when the homotopy pullback that defines the homotopy fiber is modeled, two things happen: first is replaced by , then an ordinary pullback of is computed. This turns out to remember the cocycles with values in , which are bundles with connection.
For the next step towards the nonabelian picture that is under full control, let’s look at th morphism
This is one part of the morphism in question
which I am in the process of fixing a gap in. So let’s for the moment see if we can agree on how that simpler part works.
Again, the morphism is modeled by a span
and we compute an ordinary pullback of the induced
This produces a simplicial set whose
objects are pairs consisting of a -bundle with connection ;
and a trivializatioin of the Pontryagin form ;
morphisms are are morphism of buundles with shifts of connections such that the Chern-Simons fom of the shift is compatible with the corresponding two trivializations.
This is the coarse version of the differential string structure, where only the trivialization of the real part of the degree 4 cohomology appears, not that of its integral part. To the extent that your worries already apply to this case, I suggest we concentrate on discussing this for a bit, while I continue working out the details of the more refined morphism
Okay?
*
(am on train, bad connection, sorry for typos, fixed now som in re-edit of above)
first is replaced by
Ok, this seems to be (done at an hidden level) exactly that I go on talking about. so I guess our poits of view will converge soon :)
need some sleep now, and to study the details you wrote. I’ll reply tomorrow morning.
I fixed some sort of brackets typoi in the formula in the proof in the entry. There is a Q to Urs in 50 above (not too important for now).
Jim Stasheff has had some problems in posting his comment on the mini-discussion above Urs and I had about the secondary characteristic classes. Here is his comment delivered to me by email – the rest of the entry is his.
@zskoda # 7 The standard meaning of secondary (char classes or coh ops or even two centuries ago invaraints of algebraic forms) is of something that is not defined in general but only when some relation holds among primaries e.g Massey ops secondary coh ops
for char classes, curvature vanishing is an extreme example but what is more usual cf Chern-Simons is a char class that vanishes for dimensional reasons on a class of manifolds
@Urs, # 9 if you are going to use refinement, at least make it clear from the get go that you mean a representative of a class e.g. a diff form rep
OK you say it a few lines later
but then you speak of refintin to a class in H^N_diff - what does that mean? @Urs, #11 the business of secondary classes (see above) is NOT about the connection being closed
@Urs, #12 classification of characteristic classes means e,g, H(BG) is a polynomial algebra on …?
and what do you mean by going beyond that - secondary? again see above oh, perhaps you really have in mind calculating H(BG) as one goes up the Whitehead tower?
@zskoda, #15 if I understand you, yes, ‘ordinary’ secondary classes are from the 1-group pov
the comment about - usually the tangent bundle - might could be true in physics but in math it hasn’t been before my birth certificat = PhD diss
any neglect of the fact that the class doed not depend on the connection is in the exposition and may have been fixed already
many classical theorems about the Weil algebra are not directly using the linear reps but rather compact fin dim G - if I recall
@Urs, #17 sorry to repeat but secondary class does NOT mean curvature vanishes far from it - cf Chern-Simons
classical Chern-Weil was phrased in terms of Lie groups but the dg algebra makes sense in rational (or real) homotopy theory e.g. for G jsut a based loop space
you are right about e.g. 1/6 p_2 but somewhat with the wrong emphasis p_2 is for M as just a smooth manifold, IF M is string, the the disibilty follows at the Whitehead tower level - no need to invoke infty-C-W
@ domenico_firorenza, #18 Thanks for saying ‘isomorphism classes of vector bundles’ often not said in other posts
@ urs, #19 H_dR I think I recognize but what are H_diff versus H
The latter.. explicate please
@urs #24 what is THE universal bundle? and is the existence of a universal connection easier for U(1)? the general constructions are unpleasant
was pseudo-connection defined and I mnissed it?
no angular brackets visible in my printout
what diff coh is all about - namely? maybe that’s why I’m struggling with it
@Urs, #24 I think you can say THE universal bundle meaning up to iso! how do you have existence without choosing? what’s the universal construction ?
@urs, # 33 pullback of the point!! as Zoran #38 points out later - a point as a G-space up to homotopy or some such to make it an honest statement
@urs, # 34 ‘not equivalent ther’ - what’s the equivalence there?
@urs, 39 the distinctions between E and B and P;E–> B are treated rather casually here also EG –> BG is treated as a fibration - not a bundle - when resolving * –> BG
‘not a universal bundle in the usual sense ’ sure, since not a bundle or did your mean something else?
G-action up to coherent homtopy - sure but refernce? folklore or ???
@ domenico_fiorenzo #43 groupoid of (ISOMORPHISM CLASSES 0F) principal G-bundles as you acknowledge by the end of the parag
forgetful maps - forgetting …??
Ok, this seems to be (done at an hidden level) exactly that I go on talking about.
Essentially, yes, up to the following discussion: what I write is the coefficient for cocycles that encode -bundles with pseudo -connection. This is like bundles with connection, but minus some of the usual conditions to be satisfied by the connection forms. This is what makes a resolution of : up to equivalence, every pseudo-connection can be gauged away.
BUT, nevertheless, genuine connections do appear in the homotopy fiber: that’s because we compute the ordinary pullback of
over each connected component on the right. And as discussed above, up to equivalence it does not matter which representative in each connected component we choose. So we will choose a nice representative: namley the oo-groupoid on the rights is the hypercohomology of with coefficients in the complex of sheaves
One checks that (for , see the discussion at circle n-bundles with connection for the special story at ) that this cohomology is just odinary de Rham cohomology: every cocycle is equivalent to one represented by a globalled defined closed -form.
So we may choose without restriction to pull back these along the above map. And the magic is: this picks among all pseudo-connections the genuine connections! This is because the curvature of a pseudo-connection is not a globally-defined form, but just a cocycle in the above hypercohomology. So pulling back the genuine curvature forms picks the genuine connections.
Kind of interesting how this works, right?
I fixed some sort of brackets typoi in the formula in the proof in the entry. There is a Q to Urs in 50 above (not too important for now).
Thanks for editing some typos, Zoran! I wanted to go back and work on it when I saw you had locked the entry. But that was good, because I had to get out of the train and walk home anyway!
I now have some more minutes to polish a bit more.
what I write is the coefficient for cocycles that encode -bundles with pseudo-connection
Fine. I’m ready to leave connections for pseudoconnection as the latter works better!
Still something I have to make my mind clear about: given a diagram of groupoids, what is its -pullback? I thought it was the usual fibred productof groupoids, i.e., the groupoid whose objects are triples , with an object in , an object in , and an isomorphism in between and (and whose morphism are…). but there must be something I’m missing here.
Concerning the fibered product: yes, that’s right.
Precisely, if all our obects are fibrant, then the homotopy pullback of is the ordinary limit over
.
Equivalently, for any fibration replacement of , it is the ordinary pullback of .
That’s one half of the subtlety. The other half is that a morphism of parameterized groupoids (stacks) is in general modeled by a span of ordinary morphisms .
A result of this is that when we see a morphism like in the -category, it may be much richer than an ordinary morphism of the groupoids of the same name. That’s where the connection-information gets into the game here: in order to model the morphism , one needs a span, that first lifts to -bundles with pseudo-connection, and only then maps to the curvatures: is modeled by the “anafunctor”
Domenico,
my little niece and nephew took care that I didn’t have too much time for our discussion here over the weekend, but now looking back at wat we said above I feel that maybe I should highlight the following points more, for clarification:
for every -Lie group with -Lie algebra there is an object that I write and which is such that cocycles with values in it are genuine -connections on -principal -bundles. In terms of the model, this is a subobject
of that other object that I decided to call , and which is such that cocycles with values in it may be tought of as pseudo-connections on -bundles. The inclusion picks among all pseudo-connections the genuine connections.
So for any we could simply look at
and this would be the -groupoid of -bundles with connections.
I am saying this in case my above replies gave the impression that I did not know how to encode that -groupoid. The problem is not writing this down, but understanding its intrinsic, model-independent meaning. Because a priori is something built “by hand” in the model, with no intrinsic origin in the -topos.
Except in the abelian case: for abelian coefficient the proof at circle n-bundle with connection shows that represents the intrinsically defined differential cohomology. At least over a smooth manifolds! The situation is more complex on orbifolds, where genuine -connections are actually the wrong thing to look at ! This is in fact another way to see that is “not intrinsic”.
In this language, the -Lie algebraic constructions in my articles with Hisham Sati and Jim Stasheff amount to constructing for a given -Lie algebra cocycle on that integrates to a cocycle a fibration
that lifts this to differential coefficients, and to identify the (ordinary) fibers of this morphism.
Postcomposed with the fibration these fibers compute the homotopy fibers of the morphisms
that send -bundles to the given characteristic class in real/de Rham cohomology.
Precomposed with the inclusion of genuine connections, this lifts to genuine connections
That said, I think you are right about your suggestion above to characterize an object as the (homotopy) pullback of
By what I said above, this is modeled by an ordinary pullback that produces pseudo-connections on -principal bundles that are subject to the constraint that their curvature characteristic form corresponding to behaves like that of an ordinary connection. That’s like the approximation to as seen by the class .
Between yesterday and today there has been some behind the scenes email work between Urs and me, the relevant parts of which I’m now posting on the nForum.
Domenico:
The (oo,1)-topoi perspective offers beautyful insghts on the classical theory. To me, the most striking one is the fact that connections do not have an intrinsic meaning, while pseudo-connections do. Such a statement, and all its consequences, should have a crystal clear meaning even in a purely differerential geometry framework. But since I’m not sure about the answer to the question “What is a pseudo-connection? (classically)” I see I’m loosing a lot of the picture. I’m convinced an answer to the question, for the Lie group U(1) is contained in the “Observations” in this nLab page.
But, if so, then the data of the are redundant, since is completely determined by , and . So the only part left would be the cocycle-type equation . But now, if one replaces to , one finds just and no condition on the . So I’m lost.
Urs:
Yes, this redundancy is what makes the object of pseudo-connections a resolution of the space of bundles without connection.
One way to think of it is that the are precisely the failure of the chosen forms to satisfy their connection cocycle condition. They absorb but record the mistakes we made in putting a connection on a bundle.
So the only part left would be the cocycle-type equation . But now, if one replaces to , one finds just and no condition on the . So I’m lost.
Yes, so this is essentially the proof that the canonical morphism is a weak equivalence. Intentionally, is just a trivially “puffed up” version of . A cocycle with values in involves a choice of a bunch of forms, but is made such that any choice works and nothing intrinsic has actually been chosen.
BUT the point of all this is that out of and not out of the equivalent we have a morphism to a sheaf of curvature forms
This morphism needs that we have chosen forms before, because this morphism takes these forms, and computes their curvature. If the forms were chosen such that the were non-vanishing (pseudo-connections) then the curvatures computed from them will also not be globally defined forms. But they will always be equivalent to globally defined forms. So we may compute the homotopy fibers of curv over the globally defined forms. And THIS picks among all the pseudo-connections those for which vanishes, which are the genuine connections.
So this whole business of pseudo-connections and is a way to compute homotopy fibers. The fact that the construction starts out looking very tautological and empty is the fact that the construction starts out with building a resolution which allows to model the morphism in question in the first place. Resolutions are precisely empty thickenings of something else.
At the page you link to I try to give an example that should be illustrative: notice that there is in particular the pseudo-connection for which all the are identically 0! Since they need not satisfy any condition, for a pseudo-connection we can just choose this trivial solution. Now the general nonsense says that this very “exotic” pseudo-connection with vanishing connection forms must be equivalent to a genuine connection with equivalent curvature. When one computes what this genuine connection associated to the “canonical pseudo connection” is, one finds that it is precisely the canonical connection constructed from a Cech cocycle
that one finds in the textbooks in the proof that every bundle admits a connection!
In summary, to compute the curvature characteristic classes of a cocycle of a -bundle, one has to first lift it through to a cocycle (containing no new information, but a bunch of arbitrary choices) and then compose to get
Now: a genuine conneciton is a particularly well-adapted such choice for a lift of . It is a good representative of something more intrinsically defined.
Domenico:
notice that there is in particular the pseudo-connection for which all the A_i are identically 0. […] one finds that it is precisely the canonical connection constructed from a Cech cocycle.
This is funny: it is precisely what I was thinking to: if there’s no condition on the ’s, then I can choose all of them to be zero, and then the pseudo-connection machine will provide a cohomology class only depending on the bundle (since the only data now are the ’s).. but I had no time to work out the details: you’ve been quicker in replying :)
Urs:
the pseudo-connection machine will provide a cohomology class only depending on the bundle (since the only data now are the ’s)..
Yes. Just for emphasis: this is not supposed to be a surprise: we know that the cohomology class of the curvature only depends on the bundle, being the image in de Rham cohomology of some characteristic class.
Domenico:
Well, this actually deserves still more emphasis! look at it this way: taking the trivial connection (i.e., taking the de rham differential) on each open subset is precisely the starting point when one classically defines connections in a differential geometry textbook. Next one sees that things go wrong on double overlaps and cures this by means of partitions of unit to get a wholly defined connection. Then one goes from connections to curvature forms,and finally to curvature classes. But doing this one misses the other natural path! that is, take the local data (which do not glue together) and use them to define a Cech cohomology class. That’s the curvature class from scratch!
Urs:
Yes, I guess you are saying that it is useful to think of non-genuine pseudo-connections in the first place.
In fact I think this point is widely under-appreciated. I originally became alerteded of this in Hamburg, when we were studying equivariant gerbes with connection: it comes initially as a surprise to see that taking the 2-stack of gerbes with connection and then picking equivariant objects in there as usual gives the wrong objects! It misses plenty of objects that should be there. The trouble is the equivariance condition on the genuine connections on the gerbes, this turns out to be much too strict.
And the solution is: if you have an orbifold then it is not correct to think of its differential cohomology as morphisms . Rather you should play that homotopy-pullback game for . When one forms that homotopy homotopy pullback, one obtains first of all the right objects, and second one notices that these contain some objects that are only equivariant as gerbes with pseudo-connection.
One way to think of this is: genuine connections are well-adapted nice representative of pseudo-connections – but only over manifolds. More generally this nice choice does not exist.
Domenico:
I was also thinking that the datum of a -pseudoconnection is precisely the datum of a collection of local -connections (relative to a good open cover with an associated trivialization of the bundle). Since we are considering a trivialized bundle on each , in order to have a notion of pseudoconnection we do not actually need the full notion of a connection (that would be quite recursive…), but only of a “connection on a trivial bundle”.
These have a natural functorial interpretation: if is a trivial principal -bundle ( a Lie group), then a -connection on is nothing but a smooth functor . Now we have a local association and the natural question is: how does this globalize? the natural answer I can imagine is: since the association above is clearly contravariant, it is a prestack over Cartesian spaces (just a fancy way of restating we are working with good open covers). So far saying “prestack” adds nothing to our picture, it is just a translation from differential geometrish to stackish. But once we have done this translation, we see there is now a natural way to globalize: stackify! And this, I strongly suspect, will produce pseudoconnections rather than connections. Namely, to give a “global section” of the stack thus obtained we do not have to give local connections which “coincide” on the overlaps, but local connections and local isomorphisms on the overlaps. Moreover, these local isomorphisms will have to satisfy a compatibility condition (a cocycle condition) on triple overlaps. So here it is: to a “natually globalized connection object” it corresponds a Cech cocycle with coefficients in local isomorphisms between local connections.
So far everything seems clear and neat, but actually actually, here we are still missing a main ingredient: curvature! curvature tells us what happens when our paths in move and sweep 2-dimensional surfaces. So we get into 2-categorical business. And it is precisely this looking at an higher categorical level that simplifies it all: namely, the groupoid is promoted to , a fancy way of saying that the local pair (connection, curvature) is better (it is flat!) than connection alone. The interesting fact is that (at least classically), the connection uniquely determines the curvature, so we can just write “connection” to mean the pair. But this should not distract us from the “real” object, that is the pair (connection, curvature). It’s a bit like in linear algebra: once basis are fixed, a linear morphism is the same thing as a matrix, and we can do a lot of operation with matrices; but we should not think of the matrix as the real object: the morphism is the real thing!
So the real prestack we are dealing with should rather be . Then we stackify and this time we should have the correct thing. This should be the (oo,1)-topos theoretic background.
Now that we have been detouring into it, let us translate back to differential geometrish. Once we’ve seen our local connections as functors, we have no other natural choice for local isomorphisms than natural isomorphisms of functors. At the 1-functors level, these turn out to be nothing but local gauge transformations. But at the 2-functor level we see another totally classical feature of connection arising: the space of connection is an affine space over the space of 1-forms. So any connection can be shifted by a 1-form, and our natural 2-morphisms between two local connections (seen as functors ) will be precisely these data: a local gauge transformation and a local shift such that the second connection is obtained by shifting the gauge-transformed first connection.
Since connections are an affine space over 1-forms, there is exactly one shift on each inducing the isomorphism between the local connection from and the gauge transformed one from . This means the addititional datum of local shift is actually redundant; nertheless, it is important to think of this datum, since as we’ve seen above it is an essential part of the real object: the local natural 2-isomorphism. And it will be the real object that matters in defining the curvature class.
Urs:
Now we have a local association and […] local connections and local isomorphisms on the overlaps.
Yes, I essentially agree with this, but I think the following subtleties are important:
The prestack that sends a manifold to the groupoid of smooth functors from the smooth path 1-groupoid to the one-object groupoid of the Lie group is ideed the prestack of trivial -bundles with connections. It’s stackificaiton is the stack of general -bundles with genuine connection.
(Not with pseudo-connections! The isomorphism on the double intersections are iso of bundles with connection and hence give precisely the cocycle condition on the local connection forms).
To get pseudo-connections in this language, we consider the prestack . This assigns to the groupoid whose objects are trivial -bundles with connection, but whose morphisms are morphisms of trivial -bundles without necessarily respect for the connection. This prestack is what I’d call . It is equivalent to just . This is an important subtlety: the collection of pseudoconnection in itself carries no information, it is a thicker groupoid equivalent to . The point is rather that inside this thicker groupoid we may “carve out” the groupoids of bundles with connection, which however we cannot “carve out” of BG itself.
Domenico:
To get pseudo-connections in this language, we consider the prestack
Yes, that’s what I say in the second part of my email (which in turn is a version of what you wrote on nLab..). First part was a first order approximation :)
Urs:
Right. By the way, I said this somewhat incorrectly just now. Just for the record, let me say it once more even: since is actually just with a group structure, it is contractible. Hence so is . The object is in fact the strict pullback of along the inclusion . That is in fact another way of seeing that is equivalent to : it is the pullback of along a map between two contractible objects.
Wait a minute! Are -pseudoconnections the same thing as flat -connections?
Wait a minute! Are -pseudoconnections the same thing as flat -connections?
On an ordinary -bundle, yes.
Where “on an ordinary -bundle” means: if you restrict the flat differential -cocycle
along the canonical inclusions to its underlying bare cocycle
then this 2-group cocycle must be assumed to factor through the inclusion and hence encode an ordinary -bundle.
Yes! I’m always thinking of ordinary -bundles at the moment: it seems the intrinsic (oo,1)-topos perspective is sheding a beautiful light on the classical theory of connections and curvature, and are these classical aspects I am most interested in at the moment. So far I have a neat understanding of connections and pseudoconnections in terms of global sections of extremely natural and simply defined stacks. And also the feeling that the natural path is not connections –> curvature, but pseudoconnections –> curvature. But now I have to make this feeling something concrete, so now I have to go and understand the business :)
By the way, one more comment on the above:
one can think of this in terms of the two conditions on an Ehresmann connection.
An Ehresmann conneciton is, as you know, g-vlaued form, satisfying two conditions:
on the fibers it restricts to the canonical form
it is equivariant.
We may think of the form as an element in .
The first Ehresmann condition corresponds to the condition that restricted to points, this becomes the cocycle of a G-bundle.
The second Ehresmann condition corresponds to the requirement that the curvature forms descend, and hence that this pseudo-connection is really a connection.
Maybe I finally see some light (or I am Urs-possessed, which is hopefully the same thing..).
As far as concerns the classical theory (-principal bundles, with a Lie group) the first step is: forget everything you know about principal bundles, connections, curvature, etc. The second step is: consider a few extremely simple functors defined on differential manifolds, e.g. , and the natural morphisms between these functors (also these morphisms are extremely natural, so much to appear quite trivial at times). The thirs step is: look at these functors as prestacks on the site of differential manifolds, and consider the associated stacks; morphisms between the original prestacks are now promoted to morphisms of stacks. The fourth and final step is: have a close look to these stacks, and see they are nothing but very clasical objects and constructions: principal bundles, connections, curvatures…
There is a further step in this story: where did we use the fact we were on the site of differential manifolds? but this is another story (and Urs is telling it).
You need things like the fact smooth manifolds are locally contractible, and have enough good covers, but apart from this, it’s all very general, as Stel’s master thesis (there’s a page on Urs’ web for it), shows.
it’s all very general
Yes, that’s indeed true. One can set this up in a rather mind-blowing generality, once one sees what the abstract pattern is in the familiar case.
But specifically, what is important about the -topos over the site CartSp or ThCartSp (the -Cahiers topos) is that for this one the abstract definitions do reproduce the familiar smooth differential geometry – as opposed to something more exotic but similarly-behaved.
For instance there is for every groupal object in a -connected -topos the object that plays the role of the coefficient for -valued generalized de Rham cohomology. Speicifically in the topos over , this happens to be given by the familiar complex of sheaves of smooth differential forms.
Dually the same stament holds for the path -groupoid: it exists in great generality, but over it is actually modeled by the familia smooth paths.
Speicifically in the topos over CartSp, this happens to be given by the familiar complex of sheaves of smooth differential forms.
And specifically this is what I’m aiming to at the moment: ordinary differential geometry of principal bundles offers a perfect palyground where can make the abstract ideas play while remaining concrete. It is a very concrete and familiar example, but at the same time one perceives the whole abstract structure in the background. A slogan for this could be: Connections and curvature: an exercise in stackification.
let me try to collect all the local (i.e. prestack) ingeredients of the construction to see if I got it (at least within some approximation degree). I will remain classical, so will be a Lie group in what follows and the context will be that of classical differential geometry. writing for the prestack , and saying “leading” to mean “which upons stackification produces”, the involved prestacks should be:
, leading to principal -bundles
, leading to princiapal -bundles with connections
, leading to principal -bundles with pseudoconnections (this includes the curvature data)
leading to flat --gerbes (with connection)
, leading to --gerbes
the “transition” from -bundles to -gerbes is via the hom-space
classically corresponding to invariant polynomials.
Yes.
Except maybe this one point: just to remove all possibilities for misunderstandings, I want to emphasize the following once more:
is equivalent to the point. By itself encodes flat -2-bundles, but each of these is equivalent to the trivial one, in this 2-groupoid.
The role of is not by itself, but that using this one can build a resolution of that encodes pseudoconnections.
Then
is bundles with pseudo-connection.
And another comment: the bit about invariant polynomials is essentially right, but maybe it is a bit more subtle:
One simple statement that is true is that every invariant polynomial gives a functor
where on the right we have the sheaf of 1-groupoids whose objects are closed -forms and whose 1-morphisms are -forms modulo exact forms.
Recall, this is as in Simons-Sullivan structured bundles.
this functor sends a connection to its curvature characteristic form and a shift between two connections to the corresponding Chern-Simons form, which is indeed well-defined modulo exact forms.
If one wants to get rid of this truncation, one has to be a bit careful with some global issues. The simple construction goes through untruncated when one is high enough in the Whitehead tower.
One more comment: notice that this pullback sheaf
is the sheaf that assigns to the groupoid whose objects are diagrams of this form
and whose morphisms are homotopies of such diagrams (compatible homotopies of the top and bottom morphism).
the truncation issue I have to understand better. at the differential forms level it seems quite clear how the pattern should continue: should have closed -forms as objects, -forms as 1-morphisms, and -forms modulo exact forms as 2-morphisms. so the untruncated version suggested by this should be the de Rahm chain complex arising in Deligne cohomology. In particular, stackifying should produce cocycles in Cech-de Rahm hypercohomology, which seem a natural way to obtain ordinary cohomology classes in the end. but you are warning me that this is a too naive picture, so I have to understand it better.
Link to Simons-Sullivan structured bundles is not working.
it seems quite clear how the pattern should continue
Well, of course it is clear what the untruncated is: it is a theorem that this is the image under Dold-Kan of the complex of sheaves .
But the thing is that for invariant polynomials of degree greater than two, the functor does not exist:
because on the left morphisms are shifts of connections and , and they compose strictly. They are sent to Chern-Simons-Forms, but these don’t quite compose strictly in general
So there is a subtlety here.
One way to resolve this is to to further resolver . For instance we can always regard this 1-groupoid as an equivalent 2-groupoid whose morphisms are paths in and whose 2-morphisms surfaces in .
Accordingly, then morphsims in are actual paths in the space of connections, and so we retain a bit more information on precisely which CS-form to choose.
One can show that is suffiicient for producing the first Pontryagin degree 4 class as a functor
to the untruncated de-Rham coefficient object.
But to go to the second degree 8 Pontryagin class, one has to be more sophiscticated. One way is if the first Pontryagin class vanishes, then we have a lift through and using that we can replace by a 7-groupoid out of which we do have a functor to that gives the second Pontryagin class.
I try to discuss this in a bit more detail on the oo-CW theory page. But the discussion needs more polishing.
Simons-Sullivan structured bundles is not working.
Google sees it. Try this one Simons-Sullivan structured bundle
ah, ok. so in my “list of ingredients” above I have to add to write the cospan
,
right?
Right, that’s a way to put it.
There is one gap in my understanding:
I do know the pretty evident way to construct these morphisms into the truncated object for each degree invariant polynomial.
I also do know then, using Simons-Sullivan, that the ordinary pullback of curvature classes along the induced
(where runs over generators of the collection of invariant polynomials) is a groupoid whose connected components are Simons-Sullivan structured bundles. Hence that this gives the “unstable” version of differential K-theory, by their central result (i.e. itt gives differential K-theory after Grothendieck-group completion).
So this sounds good. But right now my trouble is that I am lacking an explanation of the abstract -categorical origin of these constructions. That ordinary pullback I mention above must model some homotopy pullback. But I don’t yet see which one.
So this is something that still bothers me.
in #77 I’m not too surprised of Chern-Simons forms not compose strictly: a relation of the form
is precisely what I would expect. I would even call a ternary characteristic class.
Yes, right. Not sure how you mean your comment, but just to clarify: I didn’t mean to say that it is surprising, just that it is a fact that shows why this truncation-business comes in.
But I think you hit the nail on the head with the term “ternary characteristic class”. That gives the right picture of precisely the issue we are confronted with here.
in #77 I’m not too surprised of Chern-Simons forms not compose strictly
evidence that CS(-,-) is a pseudofunctor or similar?
evidence that CS(-,-) is a pseudofunctor or similar?
Yes, that’s effectively what we have been discussing: for a simply connected Lie group the degree 4-classes can be realized as 4-functors that send
a connection to its Pontryagin form
a path of connections to the associated Chern-Simons form
etc. It is due to the fact that every Lie group has vanishing that this is actually coherent.
For the same reason this straightforward assignment fails for higher classes. One way to fix this is to pass to higher covers of . Out of the String-2–group there is indeed an 8-functor
which sends a connection to its second Pontryagin form . This is effectively because has vanishing .
This is explained in some detail at infinity-Chern-Weil theory. I plan today to polish that entry further.
@ Urs: in #75 you mean ?
Domenico,
yes. That’s the same!
Except maybe that strictly speaking is what I use for the abstract -functor and here when we write this pullback it is all about working with very specificic models so that one should say which model precisely for we mean. We should mean the model , where the hom is that of diffeological 2-groupoids.
Domenico and David,
I started writing out details of the Chern-Simons form business that we have been discussing in the last messages at Chern-Simons form in the new section In oo-Chern-Weil theory.
It breaks off at some point. Because I have to run to grab some breakfast now!
it is all about working with very specificic models
Maybe I should expand on this:
the simplest model for is simply the constant presheaf, that is constant on the bare -groupoid that underlies .
But for our purposes, of central importance is the de Rham coefficient object which is the homotopy fiber of the canonical morphism .
To handle that conveniently, we want a bigger model of such that this morphism becomes a fibration.
That’s where differential forms come in. Their complex provides a resolution of the constant sheaf.
This is, apart from some generalization to nonabelian setting, the basic step of the de Rham theorem (as described there).
Have now added to Chern-Simons form HERE more details on how these come about from the general abstract point of view
so, the correct list of ingredients should be:
(principal -bundles)
(principal -bundles with connections)
(principal -bundles with pseudoconnections)
(de Rahm coefficient object)
and, finally, the morphism induced by an invariant polynomial.
If no ingredient is missing, in a follow up post I’ll try to display the diagram relating all these ingredients.
now I finally went through the Motivation-section and polished a bit. Notably I corrected the groupoid degrees.
Also I made all the Chern-Weil morphisms go , adding the . Domenico was quite right that this is the construction I actually give, not out of itself as I originally stated incorrectly.
Domenico was quite right..
Glad to hear this :) I apologize for having been so insistent (and maybe annoying) on that.
maybe annoying
no, not at all! I am glad we talked about this and that you caught that. That’s the point of discussing stuff here, so that it gets ironed out.
no, not at all!
:)
as promised, let me start collecting here the morphisms relating the various ingredients (this is just for the occasional reader’s convenience: all these morphisms are described in nLab, but spread across several pages).
. this is the "forget the connection morphism"; it is induced by the inclusion of constant paths
. every principal -bundle induces a principal -bundle. this is induced by the natural morphism
. this is the "forget you knew the connection was flat" morphism; it is induced by the composite morphism
. this is the "every connection is flat" principle. a clear way of looking at it is as a manifestation of the freeness property of the Weil algebra of a Lie algebra.
. this is "every connection is a pseudoconnection". it is induced by the two morphisms and described above; what one is saying is simply that given a principal -bundle with connection , the principal -bundle underlying the flat -bundle associated with is nothing but the the principal -bundle associated with the principal -bundle underlying .
Enough for today. Next morphism I will describe is the crucial
while still thinking of (edit: I had forgot the essential in the previous version of this and of the above post), let me add that picking up a morphism and using the functoriality of the construction one induces from it a morphism . In particular, by precomposing with this should give the morphism considered by Urs in #91. this should also descend to a natural morphism (or at least it is unclear to me why it would not descend to).
Good point. Let me tell you what I know, and what I don’t know:
What I know is how to model the abstractly defined morphism in terms of connectiomns and curvature invaraints in the case that itself comes from an -Lie algebra cocycle.
So if comes from Lie integration (for instance a semisimple Lie algebra and ) and is an -cocycle
then we have a canonical functorial morphism
where the lattice we are dividing out of is that generated by the periods of the chosen cocycle over -balls.
By finding any invariant polynomial for , this construction may be thickened to one that includes connections, by integrating in the same fashion instead the diagram
to a morphism called then .
Then it is a theorem that the abstractly defined morphism is modeled by postcomposing this further with the projection
So that’s the construction of the “-Chern-Weil homomorphism”. It is all very functorial and natural, except that there is this issue with the coskeletal truncation.
If for instance is an ordinary Lie algebra, then is equivalent to for the simply connected Lie group integrating it and also is, but already is no longer. This instead gives something similar to the String-group cover, but with a discrete group replacing where the string group is buit from a copy of .
That’s where all this business about the truncation comes in .
But now one could look at something else, wich it seems you have in mind: it so happens that the obtained by Lie integration and truncation at as above is equivalent to that pullback
that we have been discussing. In order to form this kind of construction, all we need is the -construction. This is available in full generality whenever itself comes to us modeled by a simplicial group. This is desscribed at groupoal model for universal principal infinity-bundles.
So in this sense we can form for all simplicial smooth groups , and this construction is then functorial and in particular we form every morphism functorially a corresponding morphism .
BUT, for this construction, since it no longer works over Lie integraion, I no longer know currently how to prove that this still models Chern-Weil, i.e. that the evident composite
still is a model for the curvature characteristic class we set out to model.
From the looks of it it likely is. But I don’t have control over this construction currently.
I like your summary list above. Now started to include such a summary in infinity-Chern-Weil theory – preparatory concepts.
But, as I said in another thread, I am currently having a huge problem that I always get error messages when trying to save this particular entry ! I am at a bit of loss as to what to do about it.
So I couldn’t save it and instead tried if maybe somehow it’s the title that causes problem. So I changed the titled and saved it as infinity-Chern-Weil theory introduction.
That worked! But only once. Next I tried to re-edit the renamed entry – and again was prevented from it.
I hope I can dream up some solution to this problem soon.
I’ll now try to edit it to see if it works. by the way, in the Summary section I would have rather than .
edit: I corrected a couple of typos; that worked.
I would also add to the list of basic morphisms (delooped universal -bundle), and the consequent (Chern-Weil morphism).
Sounds good. Thanks for fixing typos.
I am currently busy with polishing the discussion at Chern-Simons 2-gerbe that I split off from the oo-Chern-Weil theory business. That is about a detailed unwinding of the construction of the differential refinement of the first Pontryagin class.