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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2020

    I have added pointer to the arXiv copy to the item

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2021

    Looking again at

    Does it offer a proof that BFhocolim[n]Δ opF(()×𝔸 n)\mathbf{B}F \coloneqq \underset{[n] \in \Delta^{op}}{hocolim} F((-) \times \mathbb{A}^n) preserves homotopy colimits in the \infty-sheaves FF?

    This is stated on the bottom of p. 2, but is it obvious?

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 6th 2021
    • (edited Jun 6th 2021)

    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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2021
    • (edited Jun 6th 2021)


    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

    1. the left adjoint preserves cofibrations,

    2. the right adjoint preserves fibrant objects.


    1. B\mathbf{B} will continue to preserve cofibrations after passage to the local model structure (because they don’t change);

    2. its right adjoint preserving fibrant objects is implied by your argument that B\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

    lim[n]Δ op[Δ smth n,()]:PSh (CartSp)PSh (CartSp) \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:

    (lim[n]Δ op[Δ smth n,limiX i])(U) lim[n]Δ op([Δ smth n,limiX i](U)) lim[n]Δ opH(U×Δ smth n,limiX i) lim[n]Δ op((limiX i)(U×Δ smth n)) lim[n]Δ op(limi(X i(U×Δ smth n))) limi(lim[n]Δ op(X i(U×Δ smth n))) limi(lim[n]Δ op[Δ smth n,X i](U)) (limi(lim[n]Δ op[Δ smth n,X i]))(U) \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

    PSh (CartSp) lim [Δ smth ,()] PSh (CartSp) Sh (CartSp) Sh (CartSp) \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.(?)

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 7th 2021

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2021

    All right, thanks. I’d have the urge to write out the proof in clean detail, maybe on the nLab page.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 8th 2021

    I can include an explicit statement in our next version, after it’s refereed.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2021
    • (edited Jun 16th 2021)

    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

    • XSmoothManifoldsySh (SmthMfds)X \in SmoothManifolds \overset{y}{\hookrightarrow} Sh_\infty(SmthMfds)

    and any

    • ASh (SmthMfds)A \in Sh_\infty(SmthMfds)

    we have (where square brackets denote internal homs):

    [X,ʃA]ʃ[X,A]. [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 USmthMfdsU \in SmthMfds:

    [X,ʃA](U) Sh (SmthMfds)(X×U,ʃA) PSh (SmthMfds)(X×U,(Ulim[n]Δ opA(U×Δ smth n))) lim[n]Δ opPSh (SmthMfds)(X×U,(UA(U×Δ smth n))) lim[n]Δ opPSh (SmthMfds)(X×U,[Δ smth n,A]) lim[n]Δ opPSh (SmthMfds)(Δ smth n×X×U,A) lim[n]Δ op([X,A](Δ smth n×U)) (ʃ[X,A])(U), \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 XX assumed to be a manifold, X×UX \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 XX more general than smooth manifolds?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2021
    • (edited Jun 16th 2021)

    [ removed ]

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 16th 2021

    The previous discussion is here:

    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


    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.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2021

    I see, thanks.

    The proof in #8 for arbitrary AA has the charming consequence that for absolutely every 𝒢Groups(SmoothGroupoids )\mathcal{G} \in Groups(SmoothGroupoids_\infty) the shape of its delooping

    B𝒢ʃB𝒢 B \mathcal{G} \;\coloneqq\; ʃ \mathbf{B} \mathcal{G}

    is a classifying space for concordance classes of 𝒢\mathcal{G}-principal \infty-bundles (over smooth manifolds).

    Moreover, for what it’s worth, the points-to-pieces transform

    𝒢PrincipalBundles X[X,B𝒢]ʃ[X,B𝒢][X,B𝒢]𝒢PrincipalBundles X conc \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 𝒢\mathcal{G}-principal bundles and nn-morphisms between them with that with nn-concordances between them.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2021

    back to the entry:

    For completeness, I went and spelled out (here) the definitions and the statement.

    diff, v6, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2021

    While I was at it, I have given the entry more of an Idea-section (now here).

    diff, v6, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2021

    have added (here) statement and proof of the “smooth Oka principle” (as per #8).

    diff, v6, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2021
    • (edited Jun 19th 2021)

    I have added (here) statement and proof that for every smooth ∞-group 𝒢\mathcal{G}, internal to smooth ∞-groupoids, the shape B𝒢B \mathcal{G} of its delooping B𝒢\mathbf{B}\mathcal{G} is a classifying space for 𝒢\mathcal{G}-principal ∞-bundles, up to concordance, over smooth manifolds XX:

    (𝒢PrinBund X) / concτ 0H(X,B𝒢). \big( \mathcal{G}PrinBund_X \big)_{/\sim_{conc}} \;\; \simeq \;\; \tau_0 \, \mathbf{H} \big( X,\, B \mathcal{G} \big) \,.

    diff, v8, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2021
    • (edited Jun 19th 2021)

    I wasted almost two days trying to generalize this statement (#15) to a classification of concordance classes of GG-equivariant principal \infty-bundles on good orbifolds XGX \!\sslash\! G, by the equivariant classifying space

    B GΓʃ((BΓ)G)ʃ(B(ΓG))(SingularSmoothGroupoids ) /(BG) 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,BΓGActions(H)X, \mathbf{B}\Gamma \,\in\, G Actions(\mathbf{H}), we have

    • (a) the GG-equivariant Γ\Gamma-principal bundles are modulated by morphisms XBΓX \longrightarrow \mathbf{B}\Gamma in GActions(H)G Actions(\mathbf{H});

    • (b) the internal hom in GActions(H)G Actions(\mathbf{H}) (the “conjugation action”) has as underlying object the internal hom in H\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.

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 19th 2021

    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?

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2021

    I was going for the proper equivariant cohomology on ʃ(XG)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 *G\ast \sslash G through the shape, which will work only for (*G)\prec(\ast \sslash G).

    Have to run now. Can try to later provide more details of the computation.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2021

    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)

    diff, v10, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 27th 2021

    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.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021

    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?

    • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 28th 2021

    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.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021

    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.)

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021

    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?

    • CommentRowNumber25.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 28th 2021

    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.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021

    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.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021
    • (edited Jul 28th 2021)

    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.

    diff, v11, current

    • CommentRowNumber28.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 28th 2021

    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.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021

    Okay, I changed “is proven” to “is claimed”.

    diff, v13, current

    • CommentRowNumber30.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 28th 2021

    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.

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021
    • (edited Jul 28th 2021)

    Ah, I misunderstood you.

    But then what you seem to be really pointing out is that in Prop. 2.4 here the site should be smooth manifolds. Sure, have changed it now.

    (It’s past my bedtime, though, I should say. Will be offline now.)

    diff, v14, current

    • CommentRowNumber32.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 28th 2021

    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.

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2021

    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 CartSpCartSp without changing the conclusion.

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2021
    • (edited Sep 6th 2021)

    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. ]

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2021

    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)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.

    • CommentRowNumber36.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 22nd 2021
    • (edited Sep 22nd 2021)

    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).

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2021
    • (edited Sep 22nd 2021)

    Yeah, therefore the last clause in #35: It works for coefficients AA with the property that there is dd such that [S d,A]A[S^{\geq d}, A] \simeq A and [ShpS d,ShpA]ShpA[Shp S^{\geq d}, Shp A] \simeq Shp A.

    So for instance A=BPU()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.

    • CommentRowNumber38.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 22nd 2021

    [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)?

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2021

    Not for every dd, it has to be chosen large enough to be above the truncation degree of the shape of AA.

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2021

    It’s really a simple idea, let me say it again, in fewer words: For faithful linear reps of the finite group GG on some vector space VV of large dimension, the spaces S(V)/GS(V)/G and BG=K(G,1)B G = K(G,1) are homotopy equivalent up to degrees smaller than the dimension of dd. Moreover, since the action in free, we have S(V)/Ghocolim(S(V)×G × )S(V)/G \;\simeq\; hocolim( S(V) \times G^{\times_\bullet} ). Now pull that hocolim out of the mapping spaces, and demand that the AA coefficients cannot detect the large S dim(V)1S^{dim(V)-1}-spheres that remain, then push the hocolim back into the mapping space.

    • CommentRowNumber41.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2021
    • (edited Oct 28th 2021)

    Just to say that I have now typed out the above proof idea in more detail.

    Under the assumption (here) that

    • Γ\Gamma is a D-topological group whose shape is truncated and braided,

    • GG is a finite group acting freely on any sphere,

    then a “smooth Oka principle” holds in SmthGrpd SmthGrpd_\infty in the following form (here)

    ʃMap(BG,BΓ)ʃMap(S n/G,BΓ)Map(ʃBG,ʃBΓ). 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 Γ\Gamma-Cech cohomology on S n/GS^n/G is concordant to data that is constant along S nS^n.

    The point then is that with the stack BG\mathbf{B}G thereby replaced by the smooth manifold S n/GS^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 Γ=PU()\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 GG-manifold XX to yield the general concordance classification of GG-equivariant Γ\Gamma-principal bundles over smooth GG-manifolds:

    ʃMap(XG,BΓ)ʃMap((X×S n)/G,BΓ)Map(ʃ(XG),ʃBΓ)H G 1(X;ʃΓ). 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.

    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2021
    • (edited Oct 29th 2021)

    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.

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2021
    • (edited Nov 4th 2021)

    So I think now on τ 0\tau_0, at least, it is readily proven from axiomatic cohesion and smooth Oka that for

    • GG a finite group with acts freely and continuously on some sphere,

    • XX a smooth GG-manifold,

    • ΓGrp(SmthGrpd )\Gamma \,\in\, Grp(SmthGrpd_\infty) 0-truncated with truncated shape whose delooping is classifying for Γ\Gamma-principal bundles;

    there is a natural bijection

    (GEquΓPrnBdl X) /τ 0ʃMap(XG,BΓ)τ 0Map(ʃ(XG),ʃBΓ)(ΓPrnBdl X G) /. \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=*X = \ast. With more work (and assuming that the shape of Γ\Gamma is also braided) it should hold without the τ 0\tau_0-truncation, but with that truncation we already get the classification result for GG-equivariant Γ\Gamma-principal bundles over smooth manifolds:

    for ΓAbCompLieGrp\Gamma \,\in\,AbCompLieGrp this recovers Lashof, Segal & May 1983; May 90, Thm. 3, Thm. 10;

    for ΓDscGrp\Gamma \,\in\,DscGrp this recovers May 90, Thm. 5; Guillou, May & Merling 2017, Thm. 5.3;

    for T nΓKT^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 Γ=PU()\Gamma \,=\, PU(\mathcal{H}) this recovers Uribe & Lück 2014, Thm. 15.17.

    • CommentRowNumber44.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2021
    • (edited Oct 31st 2021)

    In fact, the above statement on τ 0\tau_0 already implies the equivalence of the full homotopy types:


    (still under the above assumptions that “GG-singularities have resolutions” and that the shape of Γ\Gamma is truncated and that its delooping is classifying for Γ\Gamma-principal bundles)

    we have, for KK any smooth manifold (regarded as eqipped with trivial GG-action) the following sequence of natural bijections:

    τ 0Map(ʃK,ʃMap(XG,BΓ)) τ 0ʃMap(K,Map(XG,BΓ)) by smooth Oka τ 0ʃMap(K×XG,BΓ) τ 0Map(ʃ(K×XG),ʃBΓ) by equivariant smooth Oka as above τ 0Map((ʃK)×(ʃXG),ʃBΓ) τ 0Map(ʃK,Map(ʃXG,ʃBΓ)) \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 τ 0Map(ʃK,)\tau_0 Map(ʃ K ,\, -) of the canonical comparison morphism

    ʃMap(XG,BΓ)Map(ʃ(XG),ʃBΓ) 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=K =

    1. the point,

    2. any smooth sphere S dS^d, d +d \in \mathbb{N}_+,

    3. any multi-puncturing of 2\mathbb{R}^2,

    it follows (by this Prop.) that this comparison morphism is a weak homotopy equivalence.

    • CommentRowNumber45.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2021
    • (edited Oct 31st 2021)

    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 ʃΓʃ \Gamma and resolvable GG-singularities.

    • CommentRowNumber46.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2021
    • (edited Nov 4th 2021)

    Just for fun: there is yet one more item in the list of recovered results in #43:

    For T nΓKT^n \hookrightarrow \Gamma \twoheadrightarrow K any extension of a discrete group KK 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.)

    • CommentRowNumber47.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2022
    • (edited Jan 11th 2022)

    (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 KUKU, the picture should simply be:

    KU G τ(X)=π 0eshMap(XG,FredPU()) BPU() 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 SmoothGrpd_\infty, with τH G 3(X;)π 0SmoothGrp(XG,BPU())\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…

    • CommentRowNumber48.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 11th 2022

    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.

    • CommentRowNumber49.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2022
    • (edited May 23rd 2022)

    have taken the liberty of adding (here) pointer to our “Equivariant principal \infty-bundles” for applications and for the terminology smooth Oka principle

    diff, v17, current

    • CommentRowNumber50.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2022
    • (edited Aug 9th 2022)

    added pointer to

    where the “smooth Oka principle” appears, independently, as Theorem B / Cor. 6.4.8.

    diff, v18, current

    • CommentRowNumber51.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2022

    added pointer to:

    diff, v19, current

    • CommentRowNumber52.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 24th 2023

    Re #50: Corollary 6.4.8 in the cited reference is not the same as the “smooth Oka principle”, at least if you understand the latter as a statement about internal homs as opposed to mapping spaces. Indeed, if you trace the references (for the definition of a formally cofibrant object), it boils down to formula (4) in Section 1.2.2, which is a statement about mapping spaces, not internal homs, and appears already in my original paper with Dan and Pedro as Theorem 1.1.

    Re #8: I am preparing the final manuscript for my paper with Dan and Pedro, and I included an explicit statement (with proof) of the smooth Oka principle in the formulation using internal homs, see Proposition 1.4 in (soon to be on arXiv). (However, this is just a formal consequence of the formulation for mapping spaces, since both sides are R-local ∞-sheaves by construction.)

    • CommentRowNumber53.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2023

    Regarding Adrian’s thesis:

    formula (4) in Section 1.2.2, which is a statement about mapping spaces, not internal homs

    I think the underline on the left does signify the internal hom, for otherwise applying π !\pi_! to it would not type-check.

    But I see now that the notation declaration in the last bullet item on p. 113 leaves that a little ambiguous.

    I’ll check with Adrian.

    Regarding your note:

    Great, looks good!

    • CommentRowNumber54.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2023

    I have double-checked with Adrian: his “Diff̲ (,)\underline{\mathbf{Diff}}^\infty(-,-)” is meant to denote the internal hom.

    I don’t think there is anything to worry about regarding priority. His methods are sufficiently different, and when he later publishes his thesis as an independent proof, it will only amplify the relevance of your original insight and proof. I think.

    • CommentRowNumber55.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 25th 2023
    • (edited Feb 25th 2023)

    Re #53: Yes, I understood that the underlined Diff^∞ denotes internal hom from the very beginning. What I was trying to say is that the map in Clough’s definition is a morphism of spaces, whereas in your formulation in #8 it is a morphism of stacks, so at least formally it is a different statement: a weak equivalence of stacks, not just of spaces. But of course, the two variants are formally equivalent, since both stacks are R-local ∞-sheaves.

    The nLab article itself (Proposition 3.1, “smooth Oka principle”) talks about a morphism of stacks, not spaces. This is also what I now mention in Proposition 1.4, citing you for this variant and the terminology “smooth Oka principle”. Clough’s statement is the same as Theorem 1.1 (a weak equivalence of spaces) in my paper with Dan and Pedro, not Proposition 1.4 (a weak equivalence of stacks). It is also essentially the same as Proposition 2.4 in the nLab article (which phrases the same idea slightly differently), not Proposition 3.1. (So in other words, the reference to Clough should be moved to Proposition 2.4 from Proposition 3.1.)

    I only mentioned this because I was previously somewhat careless with the terminology and used “smooth Oka principle” for both statements, Proposition 2.4 (formulated using a mapping space) and Proposition 3.1. But now I noticed that the article only refers to Proposition 3.1 as “smooth Oka principle”, not 2.4.

    Maybe what I should ask is this: would you refer to the statement that Map(X,(∫A)(pt))≃∫Hom(X,A) as the smooth Oka principle? Here Map(-,-) denotes the mapping space (an ∞-groupoid), not the internal hom.

    By the way, another observation that may be worth making (not sure if it is mentioned explicitly anywhere) is that the shape of the associated ∞-sheaf of an ∞-presheaf F is the same as the shape of F itself. It is quite handy for concrete computations, since it allows you to bypass verifying that a given ∞-presheaf on cartesian spaces is an ∞-sheaf (e.g., for the Deligne complex). I included it as Proposition 13.9 in my newer paper.

    • CommentRowNumber56.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2023
    • (edited Feb 25th 2023)

    Oh, I see.

    Sure, the “nicest” version of the statement is the internal one, where it’s about stacks on both sides and we just permute the shape past the mapping construction.

    That’s the version that one will eventually want to state (and maybe prove…) in cohesive HoTT as

    A:Type,X:Mfdoka X,A:ʃ(XA)(ʃX)(ʃA). A \,\colon\, Type ,\;\;\; X \,\colon\, Mfd \;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\; oka_{X,A} \;\;\colon\;\; ʃ (X \to A) \;\;\simeq\;\; (ʃ X) \to (ʃ A) \,.

    I think this statement should get all the glory.

    On the other hand, with H\mathbf{H} cohesive also the fully faithful inclusion Disc:Grpd HDisc \colon Grpd_\infty \to \mathbf{H} will be tacitly understood (which Adrian insists in calling π *\pi^\ast), so that I am not sure if it is worth amplifying the distinction too much?

    For what it’s worth, traditional literature on the original Oka principle (such as quoted and cited in the entry) operates below the sophistication level of \infty-topos theory and what they actually write down is more of a model-theoretic presentation in topological spaces, where the step of applying shape is all implicit anyway.

    So I think in speaking of the “smooth Oka principle” here, we are in any case applying modernization to what other people have stated or are stating, granting that they didn’t or don’t quite give the statement its most natural home, which we however feel should tacitly be understood and will be understood in the hott future. Something like this. :-)

    • CommentRowNumber57.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 25th 2023

    so that I am not sure if it is worth amplifying the distinction too much?

    I agree that the distinction is not that severe and both versions imply each other formally.

    Not really trying to amplify anything here, but since the current nLab article makes a point to state both versions separately as Propositions 2.4 and 3.1, probably it also makes sense to ensure that the references given for each statement point to a statement of the same type in the literature.

    And if we agree that “smooth Oka principle” could refer to 2.4 in addition to 3.1, probably it makes sense to mention this somewhere around 2.4 too (perhaps in another equivalent bullet point for 2.4).

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