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I have renamed the entry formerly called (and still redirecting) “connection on a principal infinity-bundle” into connection on a smooth principal infinity-bundle.
I will now start with bringing that entry into shape.
In the same vein I have renamed the entry formerly titled (and still redirecting) “infinity-Chern-Weil theory” into Chern-Weil theory in Smooth∞Grpd.
This way things are set up well for when the legions of students arrive who will do all the analogous discussion in other cohesive $(\infty,1)$-toposes such as $Algebraic \infty Grpd$, $ComplexAnalytic \infty Grpd$ as well as the derived version of all of these. ;-)
I am reading Section 7.1 in the paper “L∞-algebra connections and applications to String- and Chern-Simons n-transport”.
Proposition 31 says that b^{n-1}u(1)-descent objects with respect to a given surjective submersion Y are in bijection with closed vertical n-forms on Y.
Take Y to be an open cover of X. Only the zero form is vertical.
Now Proposition 31 seems to assert that the unique zero vertical form is in bijection with descent objects with respect to the surjective submersion Y→X, i.e., there is a unique (?) descent object with respect to Y→X.
But in general, there is more than one way to glue a principal G-bundle with respect to the open cover Y→X, since different choices of transition functions need not be equivalent. What am I missing here?
What is denoted $A_{vert}$ plays the role of what for the classical notion of Cartan connections is the connection form restricted to the fibers of a principal bundle $Y \to X$. This restriction of a principal connection form to a fiber is flat. Accordingly, that Prop, 31 displays dg-homomorphisms out of $CE(...)$ (which are flat forms) not out of $W(...)$ (which are the possibly non-flat forms).
This perspective gets introduced around (15) on p. 9. If we think of the case that $Y \to X$ is a principal bundle, then that commuting diagram (15) shows the compatibility conditions on a Cartan connection 1-form: In the middle a possibly non-flat connection on the total space whose (on the bottom) invariant polynomials descent to the base and whose restriction to each fiber (at the top) is flat.
So I think Prop. 31 is correct as stated (for what that’s worth, it’s not making a deep point at all). What is admittedly misleading in this article is the terminology “descent object”, because we never got around to producing actual descent of higher bundles here, everything is local data. The actual descent data produced from this is the topic of the followup “Cech cocycles for differential characteristic classes” (arXiv:1011.4735)
The idea is that the actual $\infty$-bundles with $\infty$-connections are those produced by applying the construction (15) to the local model cases where $Y \to X$ is just $R\mathbb{R}^n \times \Delta^k \to \mathbb{R}^n$, collect this data into a simplicial presheaf on $CartSp$ as $n$ and $k$ range, and regard this as representing the classifying stack for the given $L_\infty$-connection. It’s the stackification involved/implicit in this prescription which glues local diagrams of the form (15) into one big structure which is a principal $\infty$-bundle with a kind of $L_\infty$-connection.
Re #4 (I presume “Guest” is Urs?)
What is admittedly misleading in this article is the terminology “descent object”, because we never got around to producing actual descent of higher bundles here, everything is local data.
Ah, I see. So it’s more like you are constructing the (pre)stack of (trivial) principal G-bundles on cartesian spaces here?
Yes, that was me, sorry. The nForum software has this most annoying habit of silently logging users out when it finds that they take too long to compile their comment.
And yes, the “$L_\infty$-connection” article must be read as being about what the local data of a principal $L_\infty$-connection ought to be, in a form that can be fed into $\infty$-stackification to produce the actual thing.
I would write it differently if I were to re-write it today. The single punchline is really that diagram (15) corresponding to a fixed $L_\infty$-algebras (which gets variously repeated, e.g. with more commentary on p. 10) and the systems of these diagrams that are induced from transgressive cocycles on that $L_\infty$-algebra (which is where the idea originated in 2007 as seen in this scan). The point is that $\infty$-stackification that local data yields the “Cech cocycles for differential characteristic classes” that give the title to arXiv.
It may help to look at the “forgotten” article by Brylinski & McLaughlin on whose title this is playing:
(I say “forgotten” because there are nearby articles by these authors that get cited a lot, but this one many people don’t seem to be actively aware of, and yet that’s maybe the key one.)
It was an epiphany I had in realizing that what Brylinski-McLaughlin do in this article is essentially the (3-truncation of) the $\infty$-stackification of that morphism of “diagrams (15)” induced from the canonical 4-cocycle on the string Lie 2-algebra.
In the description of cocycle data for principal n-bundles over a manifold $X$ covered by $\{U_i\}$, what is known/available in the literature about the relationship between differential forms on $U_{i_1,...,i_k} \times \Delta^k$, and forms on just the Čech nerve $U_{i_1,...,i_k}$ (all with values in the appropriate Lie algebras/groups) for $1\leq k\leq n$?
For example, Rist–Saemann–Wolf in their paper https://arxiv.org/abs/2203.00092 give an explicit description of cocycles for principal 2-bundles over a crossed module of Lie groups in (2.23a) and (2.23b), which uses forms over $U^{[k]}$, the $k$-fold fiber product of the open cover $U\to X$.
On the other hand, your paper also gives a description of cocycles, but using differential forms over $U^{[k]}\times\Delta^k$.
Is an equivalence between these two types of cocycle data written up somewhere in the literature?
I am not aware of any such comparison.
I had the following questions about the stacks $\mathrm{exp}(\mathfrak{g})_{\mathrm{diff}}$, $\mathrm{exp}(\mathfrak{g})_{\mathrm{CW}}$, and $\mathrm{exp}(\mathfrak{g})_{\mathrm{conn}}$ presented on this page (all of these are defined in 1.2.9.6 in https://ncatlab.org/schreiber/files/dcct170811.pdf or in section 4 of https://arxiv.org/abs/1011.4735v2):
Is the following interpretation of the simplicial presheaves $\mathrm{exp}(\mathfrak{g})_{\mathrm{diff}}$, $\mathrm{exp}(\mathfrak{g})_{\mathrm{CW}}$, and $\mathrm{exp}(\mathfrak{g})_{\mathrm{conn}}$ well informed/correct? All three are isomorphic to some groupoid of principal G-bundles with connection, but having different morphisms. Respectively these would be the groupoids with: arbitrary morphisms (not necessarily connection preserving), morphisms that are concordances of G-bundles with connection with vanishing Chern-Simons form, morphisms which preserve connection in the usual sense.
What was the intended purpose/function of the stack $\mathrm{exp}(\mathfrak{g})_{\mathrm{CW}}$? In DCCT it doesn’t appear to be used after it is defined, and in Cech cocycles for differential characteristic classes it only seems to be brought up in remark 5.2.2 after its definition. Immediately after this remark however, one restricts their attention to back to $\mathrm{exp}(\mathfrak{g})_{\mathrm{conn}}$.
If the interpretation in question 1 is correct, are you aware of any connection between the stack $\mathrm{exp}(\mathfrak{g})_{\mathrm{CW}}$ and the construction of Simons and Sullivan that uses equivalence classes of vector bundles with connection modulo concordances with vanishing Chern-Simons form (this is https://arxiv.org/abs/0810.4935)?
Thanks!
Thanks for the message, these are good observations.
Re (1): Yes!
Re (2): Right, this is not clearly written:
On the one hand, the object $exp(\mathfrak{g})_{CW}$ is the one that is nicely motivated on abstract grounds — it uses the “obvious” diagram and already yields the desired “Cech cocycles for differential characteristic classes”. This is really the object that drives the theory.
On the other hand, it may seem like an embarrassment that this object implements the expected “second Ehresmann condition” only in a weakened form. Therefore the subobject $exp(\mathfrak{g})_{conn}$ just adds the remaining condition “by hand” and runs with that. This restriction isn’t necessary if one is not prejudiced about what a principal $\infty$-connection should be like.
Therefore your item (3) is spot-on: Yes, this could be the right perspective to appreciate $exp(\mathfrak{g})_{CW}$! But I haven’t further thought about it.
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