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I have added pointer to the arXiv copy to the item
Looking again at
Does it offer a proof that $\mathbf{B}F \coloneqq \underset{[n] \in \Delta^{op}}{hocolim} F((-) \times \mathbb{A}^n)$ preserves homotopy colimits in the $\infty$-sheaves $F$?
This is stated on the bottom of p. 2, but is it obvious?
Hi Urs,
B is constructed as a left adjoint functor in a Quillen adjunction (see the paragraph before Proposition 2.5) and we prove that B preserves weak equivalences.
Combined together, this implies the claim.
Here is a complete direct proof, extracted from the paper.
The functor B is cocontinuous: (BF)_n(S) = F(Δ^n⨯S)_n depends cocontinuously on F. This formula also implies that B preserves monomorphisms.
B preserves objectwise weak equivalences since the diagonal of a bisimplicial set preserves objectwise weak equivalences.
Thus, B is a left Quillen functor for the injective model structures and B preserves objectwise weak equivalences.
Thus, B preserves homotopy colimits of presheaves, and, in particular, sends the Čech nerve hocolim_i U_i=ČU→X of an open cover U of a manifold X to hocolim_i B(U_i) → B(X), and the map is a weak equivalence by the nerve theorem. (Here, U can be assumed to be a differentiably good open cover, so the simplest version of the nerve theorem suffices.)
Hence, B is a left Quillen functor between local injective model structures, and it preserves local weak equivalences.
Thanks.
Maybe to conclude that argument one should point to the recognition theorem here, which says that a simplicial adjunction between left proper simplicial model categories is Quillen as soon as
the left adjoint preserves cofibrations,
the right adjoint preserves fibrant objects.
Namely,
$\mathbf{B}$ will continue to preserve cofibrations after passage to the local model structure (because they don’t change);
its right adjoint preserving fibrant objects is implied by your argument that $\mathbf{B}$ sends Cech nerves to weak equivalences.
And so the recognition theorem concludes the argument.
$\,$
Alternatively, maybe we could immediately consider the $\infty$-functor on $\infty$-presheaves
$\underset{\underset{ [n] \in \Delta^{op} }{\longrightarrow}}{\lim} [\Delta^n_{smth}, (-)] \;\colon\; PSh_\infty(CartSp) \overset { } {\longrightarrow} PSh_\infty(CartSp)$That this preserves $\infty$-colimits of $\infty$-presheaves follows by a formal argument, using
(a) the fact that $\infty$-colimits are computed objectwise,
(b) the formula for the evaluation of internal-hom $\infty$-presheaves,
(c) the $\infty$-Yoneda lemma:
$\begin{aligned} \Big( \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, \big[ \Delta^n_{\mathrm{smth}}, \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, X_i \big] \Big) (U) & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, \Big( \big[ \Delta^n_{\mathrm{smth}}, \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, X_i \big] (U) \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, \mathbf{H} \big( U \times \Delta^n_{\mathrm{smth}}, \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, X_i \big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, \Big( \big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, X_i \big) \big( U \times \Delta^n_{\mathrm{smth}} \big) \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, \Big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, \big( X_i ( U \times \Delta^n_{\mathrm{smth}} ) \big) \Big) \\ & \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, \Big( \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, \big( X_i ( U \times \Delta^n_{\mathrm{smth}} ) \big) \Big) \\ & \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, \big( \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, [ \Delta^n_{\mathrm{smth}} , X_i ] (U) \big) \\ & \;\simeq\; \Big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\mathrm{lim}} \, \big( \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\lim} \, [ \Delta^n_{\mathrm{smth}} , X_i ] \big) \Big)(U) \end{aligned}$Now this $\infty$-functor is homotopical for Cech-local weak equivalences and hence descends to $\infty$-sheaves
$\array{ PSh_\infty(CartSp) & \overset{ \underset{\longrightarrow}{lim}_\bullet [\Delta^\bullet_{smth},(-)] }{\longrightarrow} & PSh_\infty(CartSp) \\ \big\downarrow && \big\downarrow \\ Sh_\infty(CartSp) &\overset{ \;\;\;\;\;\; }{\longrightarrow}& Sh_\infty(CartSp) }$To conclude along these $\infty$-lines just needs an argument now that this descended $\infty$-functor is still a left $\infty$-adjoint.(?)
Maybe to conclude that argument one should point to the recognition theorem
Yes, absolutely, this is essentially the same argument.
Concerning cartesian spaces: sure, this is also a legitimate (and quite similar) argument.
To conclude along these ∞-lines just needs an argument now that this descended ∞-functor is still a left ∞\infty-adjoint.(?)
Instead of descending it using localizations, you can simply observe that it restricts to sheaves. That is to say, it sends sheaves to sheaves. This is much easier to prove than in the case of manifolds: simply observe that the resulting presheaf is R-invariant, and R-invariant presheaves on cartesian spaces are ∞-sheaves.
All right, thanks. I’d have the urge to write out the proof in clean detail, maybe on the nLab page.
I can include an explicit statement in our next version, after it’s refereed.
That would be great if it were citable as a numbered proposition in your article!
On a related note, I have a vague memory of chatting about the impliciation of your theorem on mapping stacks, but now I forget if anyone ever made notes on this: Namely it ought to be true that for
and any
we have (where square brackets denote internal homs):
$[X, ʃA] \;\simeq\; ʃ[X, A] \,.$(a kind of smooth Oka principle)
Just for the record, a proof would be the following sequence of natural equivalences in $U \in SmthMfds$:
$\begin{aligned} [X, ʃA](U) & \;\simeq\; Sh_\infty(SmthMfds) \big( X \times U, ʃA \big) \\ & \;\simeq\; PSh_\infty(SmthMfds) \Big( X \times U, \, \big( U' \,\mapsto\, \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} A(U' \times \Delta^n_{\mathrm{smth}}) \big) \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} PSh_\infty(SmthMfds) \Big( X \times U, \, \big( U' \,\mapsto\, A(U' \times \Delta^n_{\mathrm{smth}}) \big) \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} PSh_\infty(SmthMfds) \Big( X \times U, \, [\Delta^n_{\mathrm{smth}}, A] \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} PSh_\infty(SmthMfds) \Big( \Delta^n_{\mathrm{smth}} \times X \times U, \, A \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} \big( [X,A]( \Delta^n_{\mathrm{smth}} \times U ) \big) \\ & \;\simeq\; \big( ʃ [X,A] \big)(U) \,, \end{aligned}$where the key step, besides two applications of your theorem, is the third, which uses that with $X$ assumed to be a manifold, $X \times U$ is representable so that the homotopy colimit of $\infty$-presheaves may be evaluated objectwise (with Yoneda left implicit).
Might this still hold for $X$ more general than smooth manifolds?
[ removed ]
The previous discussion is here: https://nforum.ncatlab.org/discussion/6816/the-shape-of-function-objects/
Concerning your question about more general X:
Take A = Ω^n_closed, the sheaf of closed differential n-forms.
This is a 0-truncated sheaf, so in particular, [X,A] ≅ [π_0(X),A], where π_0(X) is the sheaf of sets given by the associated sheaf of the presheaf U↦π_0(X(U)).
On the other hand, ʃA ≃ K(R,n), the nth Eilenberg–MacLane space of the reals, which is n-truncated and has a nontrivial sheaf of homotopy groups in degree n.
Thus, in the expression
[X,ʃA]≃ʃ[X,A]
the left side sees at least the first n sheaves of homotopy groups of X, whereas the right side only sees π_0(X).
So there is no hope of extending this claim to sheaves X that are not 0-truncated.
I see, thanks.
The proof in #8 for arbitrary $A$ has the charming consequence that for absolutely every $\mathcal{G} \in Groups(SmoothGroupoids_\infty)$ the shape of its delooping
$B \mathcal{G} \;\coloneqq\; ʃ \mathbf{B} \mathcal{G}$is a classifying space for concordance classes of $\mathcal{G}$-principal $\infty$-bundles (over smooth manifolds).
Moreover, for what it’s worth, the points-to-pieces transform
$\mathcal{G}PrincipalBundles_X \;\simeq\; \flat [X, \mathbf{B}\mathcal{G}] \overset{ \;\;\;\;\;\;\;\;\;\; }{\longrightarrow} ʃ [X, \mathbf{B}\mathcal{G}] \;\simeq\; [X, B \mathcal{G}] \;\simeq\; \mathcal{G}PrincipalBundles^{conc}_X$canonically compares the $\infty$-groupoid of $\mathcal{G}$-principal bundles and $n$-morphisms between them with that with $n$-concordances between them.
I have added (here) statement and proof that for every smooth ∞-group $\mathcal{G}$, internal to smooth ∞-groupoids, the shape $B \mathcal{G}$ of its delooping $\mathbf{B}\mathcal{G}$ is a classifying space for $\mathcal{G}$-principal ∞-bundles, up to concordance, over smooth manifolds $X$:
$\big( \mathcal{G}PrinBund_X \big)_{/\sim_{conc}} \;\; \simeq \;\; \tau_0 \, \mathbf{H} \big( X,\, B \mathcal{G} \big) \,.$I wasted almost two days trying to generalize this statement (#15) to a classification of concordance classes of $G$-equivariant principal $\infty$-bundles on good orbifolds $X \!\sslash\! G$, by the equivariant classifying space
$B_G \Gamma \;\coloneqq\; ʃ \prec\big( (\mathbf{B}\Gamma)\sslash G \big) \;\simeq\; ʃ \prec\big( \mathbf{B}(\Gamma \rtimes G) \big) \;\;\; \in \; \big(SingularSmoothGroupoids_\infty\big)_{/ \prec(\mathbf{B}G)}$(notation as in Proper Orbifold Cohomology).
I was trying to use that for $X, \mathbf{B}\Gamma \,\in\, G Actions(\mathbf{H})$, we have
(a) the $G$-equivariant $\Gamma$-principal bundles are modulated by morphisms $X \longrightarrow \mathbf{B}\Gamma$ in $G Actions(\mathbf{H})$;
(b) the internal hom in $G Actions(\mathbf{H})$ (the “conjugation action”) has as underlying object the internal hom in $\mathbf{H}$.
My idea was to apply the smooth Oka principle to this underlying internal hom object as in the above proof (#15) and then proceed from there, which first I thought would be straightforward. But it isn’t straightforward, and now I am worried that it may not work at all.
Re #16:
Unless I misunderstood what you wrote, in the case G=O(n) or G=GL(n), wouldn’t your conjecture imply that the (equivariant) topological K-theory of X//G can be computed as the space of maps of ∞-groupoids from ∫(X//G) to ∫B(O(n)), i.e., the Borel cohomology of X//G?
And since we know that G-equivairant topological K-theory of X cannot be computed using Borel cohomology, this would imply that the conjecture is false?
I was going for the proper equivariant cohomology on $ʃ \prec (X \sslash G)$. Carrying that orbisingularization $\prec$ around is one part of what makes the equivariant generalization of #15 not quite straightforward. But it is also what made me think it should actually work, because at one point one will needs to commute a Hom out of $\ast \sslash G$ through the shape, which will work only for $\prec(\ast \sslash G)$.
Have to run now. Can try to later provide more details of the computation.
added pointer to
which claims an alternative proof of the full statement that shape = smooth fundamental $\infty$-groupoid.
(I forget if we ever talked about this preprint before, just came across this while looking for something else)
which claims an alternative proof of the full statement that shape = smooth fundamental ∞\infty-groupoid.
This claim was retracted in Version 2, after I pointed out a mistake.
I am looking at version 2, section 3:
Def. 3.2 seems to be the correct smooth singular simplicial complex (using that diagonal of bisimplicial sets is a model for the hocolim over one of the two simplicial variables).
p. 19 claims this is left Quillen (this is p. 17 in v1, but otherwise seems identical)
Prop. 3.11 claims that it sends Cartesian spaces to the point.
If true, this implies that it’s shape.
This is confirmed by the claim of Thm. 3.14 (Cor. 3.12 in v1), which says that it’s localization onto the homotopy invariant objects.
I havenn’t looked through the proofs, but the claim is there in v2, isn’t it.
Which statement exactly do you say was removed in v2?
Re #21: The statement for cartesian site (which is what you are talking about here) and the statement for the site of smooth manifolds are two radically different statements.
For cartesian manifolds, this paper indeed gives a proof.
But I would not call it “alternative”, since we also give this proof in our paper.
In fact, for cartesian site I was under a strong impression that is also present somewhere in your DCCT, but you may know better.
But the $\infty$-toposes over the two sites are equivalent. We can choose whatever site is convenient for a given construction.
In DCCT I observed that over CartSp the $\infty$-left adjoint to the constant $\infty$-stack functor exists and takes Cartesian space to the point, etc.
But in dcct I didn’t look into making the smooth singular simplicial complex functor into a left Quillen functor.
(I looked into that before writing dcct, and old notes in this direction might still be floating around somewhere (?), but then abandoned this for the other approach.)
But maybe I am misunderstanding what you are referring to: Could you say explicilty which numbered Prop./Thm. in v1 got deleted with v2 of Severin’s preprint?
But the ∞-toposes over the two sites are equivalent. We can choose whatever site is convenient for a given construction.
They are, but in order to even claim to be working in ∞-toposes, you need to show that the smooth singular simplicial complex is an object in the ∞-topos, i.e., it satisfies the homotopy descent condition.
This condition is trivial to show for the cartesian site, but highly nontrivial for the site of manifolds.
But maybe I am misunderstanding what you are referring to: Could you say explicilty which numbered Prop./Thm. in v1 got deleted with v2 of Severin’s preprint?
Of course. In Version 1, it says right before Theorem 6.6: “We can apply these insights to give an independent proof of the following representability theorem, which we have adapted to our formalism (using cartesian spaces in place of manifolds) from [BEdBP]:”
This claim was removed in Version 2. [BEdBP] refers to my paper with Dan and Pedro, and the representability theorem says the smooth singular simplicial complex functor is representable by its value on a point, working on the site of smooth manifolds.
Oh, I see your point. In fact Severin’s smooth singular simplicical complex is just the value of yours on the point. So while in both cases we may conclude that shape is equivalently given by the $\infty$-stack constant on this point-value of the smooth singular simplicial complex, your theorem is, of course, the stronger statement that shape is already given by the full smooth singular simplicial complex, even before evaluating on the point and the re-extending.
I’ll adjust the wording in the entry in a moment. But have to have some dinner first, now.
Okay, I have made the statement of the Proposition now display four conclusions, currently labeled (3) to (6), and made explicit, in the attribution below, that S. Bunk’s proof yields (6), not the stronger (3), (4).
Also adjusted the wording in the References-section accordingly.
Let me know if you agree that this is fair now.
Looks better now.
However, I see a problem in the proof of Proposition 3.1, which is stated with sheaves on smooth manifolds, yet in Step (2) it references Proposition 2.4, which is currently formulated only for sheaves on cartesian spaces.
In its current form, Proposition 2.4 is simply not strong enough to justify Step (2) of the proof of Proposition 3.1.
You need the full strength of our theorem for sheaves on smooth manifolds, not just cartesian spaces.
Urs, I am confused by your most recent change. My remark was not about Bunk’s proof, it was about your proof of Proposition 3.1, which I think has a gap in its current form. This has nothing to do with Bunk’s proof.
Looks better! But now this claim “The particular conclusion (6) is also claimed as Bunk 2020, Prop. 3.6 with Prop. 3.11.” is definitely not true, since Bunk only uses the cartesian site, and Proposition 2.4 now uses arbitrary manifolds.
But that conclusion, as stated there, is independent of the site, since it only concerns the $\infty$-topos.
Similarly, the proof of Prop. 3.1 could be stated over the site $CartSp$ without changing the conclusion.
Coming back to #16:
The kind of equivalence that I am after there, identifying topological realization of hom-groupoids with the mapping spaces between the realization of the arguments, is claimed, for some kinds of topological groupoids, at least, in the last lines of Murayama-Shimakawa 1985 (the second arrow appearing on p. 1295 (7 of 7)).
But it’s not clear to me if the authors meant this to be obvious, or well-known, or whether they meant that they checked this, without writing up the proof.
[ edit:
I see that this remark by Murayama & Shimakawa is picked up on the bottom of p. 22 in Guillou, May & Merling 2017. There the argument is: If one already knows that the two sides of the would-be equivalence model universal bundles, then the comparisom map must be an equivalence, by uniqueness of universal bundles. ]
Coming back to #34:
In the context of equivariant bundles, it would be most useful already to have the “smooth Oka principle” for the simple special case that the domain is the delooping of a finite group.
Maybe it is useful to reduce to the case that the domain is a smooth manifold (#15) by the following trick:
Pick any faithful orthogonal rep of the finite group, tensor it with an arbitrarily large (but finite dimensional) trivial rep, and then take the domain space to be the quotient manifold of the unit sphere inside that representation space. To this “spherical space form” the smooth Oka principle is known to apply, and the homotopy groups of its shape are that of $K(G,1)$ up to arbitrary high degree.
Now if we take that degree to be higher than the truncation bound of the shape of the codomain, and higher than the dimension that Cech cohomology with coefficients in the codomain can detect, then we are getting somewhere.
Re #35: Do you mean the Oka principle on the site of cartesian smooth manifolds?
I do not think it can be true: consider the case of X=BG (G a finite group, say) and A=Vect_∇ (homotopy group completed if you want), interpreted as a simplicial presheaf.
Then [X,A] is the sheaf of G-equivariant vector bundles, and its shape computes Segal’s G-equivariant K-theory.
On the other hand, [X,∫A] computes the topological K-theory of a finite group G (Borel construction).
Yeah, therefore the last clause in #35: It works for coefficients $A$ with the property that there is $d$ such that $[S^{\geq d}, A] \simeq A$ and $[Shp S^{\geq d}, Shp A] \simeq Shp A$.
So for instance $A \,=\, \mathbf{B} PU(\mathcal{H})$ should work. And in this case we recover the result that Uribe et al. laboriously prove in BEJU 14. I think.
[S ≥d,A]≃A
I am really confused by this claim when A=BPU(ℋ).
Say, if d=1, then [S^1, BPU(H)] is just the smooth loop space of BPU(H), and how can we possibly expect it to be homotopy equivalent to BPU(H)?
Not for every $d$, it has to be chosen large enough to be above the truncation degree of the shape of $A$.
It’s really a simple idea, let me say it again, in fewer words: For faithful linear reps of the finite group $G$ on some vector space $V$ of large dimension, the spaces $S(V)/G$ and $B G = K(G,1)$ are homotopy equivalent up to degrees smaller than the dimension of $d$. Moreover, since the action in free, we have $S(V)/G \;\simeq\; hocolim( S(V) \times G^{\times_\bullet} )$. Now pull that hocolim out of the mapping spaces, and demand that the $A$ coefficients cannot detect the large $S^{dim(V)-1}$-spheres that remain, then push the hocolim back into the mapping space.
Just to say that I have now typed out the above proof idea in more detail.
Under the assumption (here) that
$\Gamma$ is a D-topological group whose shape is truncated and braided,
$G$ is a finite group acting freely on any sphere,
then a “smooth Oka principle” holds in $SmthGrpd_\infty$ in the following form (here)
$ʃ Map( \mathbf{B}G ,\, \mathbf{B}\Gamma ) \;\simeq\; ʃ Map( S^n/G ,\, \mathbf{B}\Gamma ) \;\simeq\; Map( ʃ\, \mathbf{B}G ,\, ʃ\, \mathbf{B}\Gamma ) \,.$Here the first equivalence uses the truncation condition to observe that all $\Gamma$-Cech cohomology on $S^n/G$ is concordant to data that is constant along $S^n$.
The point then is that with the stack $\mathbf{B}G$ thereby replaced by the smooth manifold $S^n/G$, the smooth Oka principle over smooth manifolds applies, with which the second equvalence above follows formally by axiomatic cohesion (and using again the truncation assumption).
In the special case that $\Gamma = PU(\mathcal{H})$ this reproduces the result of Uribe et al. 2014 (here).
I am thinking that the proof should generalize, under the same assumption as above, to any smooth $G$-manifold $X$ to yield the general concordance classification of $G$-equivariant $\Gamma$-principal bundles over smooth $G$-manifolds:
$ʃ Map( X \!\sslash\! G ,\, \mathbf{B}\Gamma ) \;\simeq\; ʃ Map\big( (X \times S^n)/G ,\, \mathbf{B}\Gamma \big) \;\simeq\; Map\big( ʃ\, (X \!\sslash\! G) ,\, ʃ\, \mathbf{B}\Gamma \big) \;\simeq\; H^1_G(X;\, ʃ\Gamma) \,.$The second step here is again a formal consequence of smooth Oka with axiomatic cohesion, and the first step should have the analogous kind of Cech cohomology proof as before. But I haven’t typed that out yet.
A suggestive wording for this proof(-strategy) would be:
The Smooth Oka principle with given coefficients generalizes from domain manifolds to those orbifolds for which the coefficients cannot distinguish an orbifold singularity from its blowup in some (high) dimension.
Next I am trying to see if we can get rid of the truncation condition by taking the colimit over the Postnikov tower of the coefficients and then arguing that this can be taken out of the second argument of the mapping stack/space. But I don’t see yet what to appeal in order to take the colimit out.
So I think now on $\tau_0$, at least, it is readily proven from axiomatic cohesion and smooth Oka that for
$G$ a finite group with acts freely and continuously on some sphere,
$X$ a smooth $G$-manifold,
$\Gamma \,\in\, Grp(SmthGrpd_\infty)$ 0-truncated with truncated shape whose delooping is classifying for $\Gamma$-principal bundles;
there is a natural bijection
$\big( G Equ \Gamma PrnBdl_X \big)_{/\sim} \;\simeq\; \tau_0 \, ʃ Map\big( X \!\sslash\! G ,\, \mathbf{B}\Gamma \big) \;\; \simeq \;\; \tau_0 \, \Map\big( ʃ( X \!\sslash\! G ) ,\, ʃ \mathbf{B}\Gamma \big) \;\simeq\; \big( \Gamma PrnBdl_{X_G} \big)_{/\sim} \,.$The proof is essentially the same as indicated above for the case where $X = \ast$. With more work (and assuming that the shape of $\Gamma$ is also braided) it should hold without the $\tau_0$-truncation, but with that truncation we already get the classification result for $G$-equivariant $\Gamma$-principal bundles over smooth manifolds:
for $\Gamma \,\in\,AbCompLieGrp$ this recovers Lashof, Segal & May 1983; May 90, Thm. 3, Thm. 10;
for $\Gamma \,\in\,DscGrp$ this recovers May 90, Thm. 5; Guillou, May & Merling 2017, Thm. 5.3;
for $T^n \hookrightarrow \Gamma \twoheadrightarrow K$ the extension of any finite (or just discrete) group by a torus this proves the conjecture in Rezk 2014, Footn. 4 ( p. 6);
for $\Gamma \,=\, PU(\mathcal{H})$ this recovers Uribe & Lück 2014, Thm. 15.17.
In fact, the above statement on $\tau_0$ already implies the equivalence of the full homotopy types:
Because
(still under the above assumptions that “$G$-singularities have resolutions” and that the shape of $\Gamma$ is truncated and that its delooping is classifying for $\Gamma$-principal bundles)
we have, for $K$ any smooth manifold (regarded as eqipped with trivial $G$-action) the following sequence of natural bijections:
$\begin{array}{lll} & \tau_0 \Map \big( ʃ K ,\, ʃ Map( X \!\sslash\! G ,\, \mathbf{B}\Gamma ) \big) \\ & \;\simeq\; \tau_0 ʃ \Map \big( K ,\, Map( X \!\sslash\! G ,\, \mathbf{B}\Gamma ) \big) & \text{by smooth Oka} \\ & \;\simeq\; \tau_0 ʃ Map( K \times X \!\sslash\! G ,\, \mathbf{B}\Gamma ) \\ & \;\simeq\; \tau_0 Map \big( ʃ ( K \times X \!\sslash\! G ) ,\, ʃ \mathbf{B}\Gamma \big) & \text{by equivariant smooth Oka as above} \\ & \;\simeq\; \tau_0 Map \big( ( ʃ K ) \times ( ʃ X \!\sslash\! G ) ,\, ʃ \mathbf{B}\Gamma \big) & \\ & \;\simeq\; \tau_0 Map \big( ʃ K ,\, Map ( ʃ X \!\sslash\! G ,\, ʃ \mathbf{B}\Gamma ) \big) & \end{array}$whose composite is the image under $\tau_0 Map(ʃ K ,\, -)$ of the canonical comparison morphism
$ʃ Map( X \!\sslash\! G ,\, \mathbf{B}\Gamma ) \xrightarrow{ \;\; } Map\big( ʃ ( X \!\sslash\! G) ,\, ʃ \mathbf{B}\Gamma \big)$Now considering these bijections for the collection of cases that $K =$
the point,
any smooth sphere $S^d$, $d \in \mathbb{N}_+$,
any multi-puncturing of $\mathbb{R}^2$,
it follows (by this Prop.) that this comparison morphism is a weak homotopy equivalence.
This finally proves, I’d think, the conjecture at the very end of Murayama & Shimakaw 1995 (raised in #34 above) for the case of truncated $ʃ \Gamma$ and resolvable $G$-singularities.
Just for fun: there is yet one more item in the list of recovered results in #43:
For $T^n \hookrightarrow \Gamma \twoheadrightarrow K$ any extension of a discrete group $K$ by a torus, the orbi-smooth Oka statement above proves the conjecture of Charles Rezk in footnote 4 on p. 6 of Global Homotopy Theory and Cohesion.
(Maybe this thread has become unintelligible now. I’ll eventually provide pointer to a pdf-writeup of the proof.)
(just to say for the record that the writeup announced in #46 is the one here)
$\,$
Moving forward, from equivariant bundles to the equivariant cohomology theories they are twisting:
Your (Dmitri’s) observation re Atiyah-Segal completion in Cor. 3.21 of the 2014 note seems spot on. Proper equivariant cohomology is a topic in cohesive homotopy theory.
I am wondering that in refining your angle there to include the 3-twist of equivariant $KU$, the picture should simply be:
$KU_G^{\tau}(X) \;\; = \;\; \pi_0 \, \esh \, Map \big( X \sslash G ,\, Fred \sslash PU(\mathcal{H}) \big)_{B PU(\mathcal{H})}$(on the right the shape of the slice mapping stack of D-topological quotient stacks formed in $SmoothGrpd_\infty$, with $\tau \,\in\, H^3_G(X; \mathbb{Z}) \simeq \pi_0 SmoothGrp( X \sslash G , B PU(\mathcal{H}) )$).
Which would be a thing of beauty if correct. Need to see to write a proof…
Re #47: Looks reasonable to me.
In fact, I’ve been meaning (once I have time) to write down a much better model for twisted K-theory (including higher twisted K-theory): start with a nice sheaf model for topological (or differential K-theory), such as super vector bundles with a superconnection, appropriately group completed, take its ⊗-invertible part (i.e., bundles of virtual dimension 1 or -1), deloop once and sheafify. This sheaf encodes all twists in higher K-theory (not just H^3(-,Z), but the whole B(GL_1(KU))) and allows for geometric models of the higher twisted Chern character etc. for twists in this generality.
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