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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.
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