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I was trying to understand Chen’s iterated integrals a bit more abstractly and ran into the following confusion about loop spaces.
Let $X$ be a smooth manifold, viewed as a simplicial presheaf over $\text{Mfld}$ or $\text{CartSp}$. It seems to me there are two different notions of the free loop space of $X$. The first is what’s described on free loop space object. This is as the powering $X^{S^1}$ or alternatively as the internal hom $[\text{LConst}(S^1), X]$. The second is the internal hom $[S^1, X]$ where here $S^1$ is the usual smooth manifold under Yoneda.
These two notions seem to be quite different — roughly, the first does not seem to remember the smooth structure on $S^1$. Is that true? (for instance the global sections seem to be different…) How are these two objects related?
Indeed, this is a crucial difference.
Notice that if by $X$ you always mean the presheaf $\mathbb{R}^n \mapsto C^\infty(\mathbb{R}^n, X)$, then in fact $[LConst(S^1), X] = X$, since there are then no non-trivial morphisms in $X$, and so the the non-trivial morphism in $LConst(S^1)$ necessarily goes to the identity on some point.
So the issue is that $X$ exists in different modes: once as a smooth manifold (which has possibly non-trivial smooth structure but only trivial morphisms), once as its shape or path infinity-groupoid $ʃ X \coloneqq LConst \Pi_\infty(X)$ (which has trivial smooth structure but possibly non-trivial morphisms).
Same for the circle $S^1$ itself. And similar then for the loop space: there is the smooth loop space $[S^1, X]$ (smooth structure, no morphisms), and the free loop space of the homotopy type $[ʃ S^1, ʃ X]$ (morphisms, no smooth structure).
The relation should be
$ʃ[S^1, X] \;\simeq\; [ʃ S^1, ʃ X]$
(by the statement in… let’s see, section 4.2 in arXiv:math/0612096)
Thanks Urs, I think I follow the idea. I agree with the computation $[\text{LConst}(S^1_\bullet), X]=X$ at the level of zero simplices but I will have to work out the higher simplices myself (the simplicial enrichment seems to involve copowering with $n$-simplices… later edit: ah never mind, it’s clear. As you said the target is discrete so all that matters is the zero-simplices of the domain. Got it!)
My motivation for asking was the following confusion in trying to understand Chen’s iterated integrals abstractly.
Chen’s iterated integrals, at least as explained by Getzler, Jones, and Petrack, can be written as a transgression-type procedure (see the top of page 13): in weight $k$,
first pullback your $(k+1)$ differential forms from $X^{k+1}$ to $LX\times \Delta^k$ under the evaluation map sending $(\ell, t_1,\ldots,t_k)\mapsto (\ell(0),\ell(t_1),\ldots,\ell(t_k))$
then integrate out the simplex directions via the projection $LX \times \Delta^k \to LX$ These fit together to yield map of chain complexes $C(\Omega^*X, \Omega^*X)\to\Omega^*(LX)$ essentially because of the families Stokes theorem.
A less ad hoc interpretation of this construction is that we are choosing a cosimplicial model for the free loop space $LX^\bullet$ coming from the standard simplicial circle (hence the $X^{k+1}$ appearing) and then studying the pullback by the evaluation map (of cosimplicial objects) $\Delta^\bullet\times \text{Tot}(LX^\bullet) \to LX^\bullet=X^{\bullet+1}$ followed by fiber integration. This is exactly how Jones sets up the corresponding map for singular cochains.
If I want to make the previous paragraph precise for differential forms I might choose to work in simplicial presheaves so that I can make sense of the free loop space and differential forms on it. But now I am a bit concerned that the loop space appearing in the construction above is only seeing $S^1$ simplicially… shouldn’t we be considering the smooth loop space $[S^1, X]$? I think I’m missing something very basic here.
the families Stokes theorem
the only reference I know for this is the first volume of Greub-Halperin-Vanstone, where it is an exercise (if I have got the right notion: so you mean Stokes for integration along the fibre of a bundle of manifolds-with-boundary?). Do you have other references? GHV use idiosyncratic notation throughout, as well as an overly complicated setup for their definition of integration over the fibre.
Do you have other references?
https://mathoverflow.net/questions/202428/reference-for-a-proof-of-the-fiberwise-stokes-theorem
Yeah, that’s what I mean. I don’t have a reference unfortunately – I’m in the same boat as Dmitri (and one should not forget that simplices have corners…).
The reason I ask is that we used the GHV version and got quantity = - quantity, for something non-zero. Clearly mis-applied, but their formalism is so odd it wasn’t clear how. Also, we were pushing along a map between infinite-dimensional manifolds with finite-dim fibres, so some pinches of salt need to applied to the literal statement of the theorem.
Re #7: Theorem 3.4.54 in Nicolaescu’s lecture notes does have different signs for two of the terms. Perhaps there is a typo in GHV.
@Dmitri, yes, it’s possible. What would be good is if there was a modern proof of the theorem spelled out, for instance on the nLab :-) :-) :-)
Re #9: Well, we could copy-paste this proof: https://math.stackexchange.com/questions/95215/stokes-for-integration-along-the-fiber#95257
Sure, but note that the signs there agree with GHV and hence not Nicolaescu. I suspect it comes down to ’fibre-first’ or ’fibre-last’ conventions around orientations. But what would be better is a version of the theorem where the map lives in the category of diffeological spaces, but where the fibres are honest manifolds with boundary. This would cover the case of Fréchet manifolds that I was considering, but also other cases of interest for people around here.
Not sure if it is of interest here, but for fiber integration over fibers of a trivial bundle, there is a neat and very general formula implicit in Gomi-Terashima 00. This is given there for fiber integration in Deligne cohomology, but as a special case this includes fiber integration of differential forms.
Re #12
aha, the signs in Theorem 3.1 of Gomi-Terashima agree with Nicolaescu, interpreted back in terms of forms.
I might at some point try to write the proof of Stokes out in detail for myself (trying to get the signs right, i.e. consistent with G&T!). If/when I do, I’ll put it up!
In the meantime, looking back at #2, I realized that I don’t actually understand the internal hom in simplicial presheaves concretely. Let’s look at the most basic example, the internal hom $[*,X]$, which I expect to be (equivalent to?) just $X$.
Here’s my attempt at computing it concretely: for $U$ a test manifold,
$[*,X](U)_n = \text{Maps}(*\times U, X)_n = \text{Hom}_\text{sPsh}(U\cdot\Delta[n], X) = \text{Hom}_\text{Psh}( (U\cdot \Delta[n])_0, X_0).$Here $\cdot$ is the copowering of simplicial presheaves over simplicial sets. But if I’m not mistaken $(U\cdot\Delta[n])_0=\coprod_{n+1} U$ as presheaves of sets. So I seem to be getting
$[*,X](U)_n = C^\infty(\coprod_{n+1}U,X)$instead of $C^\infty(U,X)$ independent of $n$. Am I misunderstanding the definition of the internal hom? Or should I be looking for an equivalence instead of an isomorphism?
EDIT: never mind, I was being very confused. I think I’ve figured it out. My question about why Chen’s map seems to use the wrong loop space still stands, though.
Re #11:
But what would be better is a version of the theorem where the map lives in the category of diffeological spaces, but where the fibres are honest manifolds with boundary. This would cover the case of Fréchet manifolds that I was considering, but also other cases of interest for people around here.
Doesn’t this follow automatically from the case of bases being ordinary manifolds? If F is a typical fiber (a smooth manifold with boundary), denote by Ω^n_F the moduli stack of bundles with a typical fiber F (dim F = k) whose total space is equipped with a differential n-form.
Then fiberwise integration yields morphisms A: Ω^n_F→Ω^{n−k} and B: Ω^n_{∂F}→Ω^{n−k+1}, as well as i*: Ω^n_F→Ω^n_{∂F}, d_F: Ω^n_F→Ω^{n+1}_F, and d: Ω^{n−k}→Ω^{n−k+1}.
Thus, we have a formula d∘A = A∘d_F ± B∘i* on the level of sheaves of abelian groups. Diffeological spaces embed into sheaves of sets fully faithfully, so taking hom from a diffeological space will give you what you want.
@Dmitri
hmm, ok. I’ll have a think, but I assume you are correct.
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