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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012
    • (edited Apr 27th 2012)

    [… I had a question here which I have answered meanwhile – I’ll post something a little later …]

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeApr 27th 2012

    The subject looked so interesting!

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012

    Yes, sorry. I kept oscillating back and forth between thinking it’s obvious and thinking that I must be missing something.

    So I am trying to characterize compact objects in Sh(SmthMfd)Sh(CartSp)Sh(SmthMfd) \simeq Sh(CartSp).

    First of all I want to say: a compact manifold (in the ordinary sense) is a compact object in there. I thought various times it’s easy to show, but then found that it’s maybe more subtle. Or I am being stupid.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeApr 29th 2012

    a compact manifold (in the ordinary sense) is a compact object in there

    You mean a closed manifold: a compact, boundaryless, finite-dimensional, locally Cartesian smooth space? And nothing else is a compact object? That would be an interesting result!

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2012
    • (edited Apr 29th 2012)

    Hi Toby,

    no, I don’t mean this and this cannot be true: every finite colimit of compact objects will be compact, and compact manifolds (with or without bounday) won’t be closed under finite colimits.

    In fact I now think that a proof that compact manifolds are compact objects here best proceeds via first establishing that closed disks are compact objects, and then build compact manifolds from gluing finitely many closed disks.

    I will try to write out details now. It all seems very easy on first sight, but there are maybe some subtleties to take care of. Also, I really want all this of course in the \infty-topos over smooth manifolds, eventually.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeApr 29th 2012

    That looks very interesting. Have you any idea what the ’extra’ compact objects might be?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2012
    • (edited Apr 29th 2012)

    I have no idea how to characterize all of them. But if I am right that the closed nn-disk is a compact object in Sh(SmthMfd)Sh(SmthMfd) then an obvious class of compact objects built from that are piecewise smooth finite CW-complexes. Things like this.

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeApr 29th 2012

    What about the orbifold models (and hence probably compact orbifolds)? I have no feeling for this, but they seem to be quotients of a disc by a finite group action, so that might work???

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2012

    True, orbifolds of compact manifolds should be compact objects in Sh (SmthMfd)Sh_\infty(SmthMfd).

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2012
    • (edited Apr 30th 2012)

    So I seem to be able to show it not for filtered diagrams, but for monofiltered diagrams. Here are some notes (intentionally at an expository level of detail):

    +– {: .num_defn #MonoFilteredDiagram}

    Definition

    Call a filtered diagram A:ICA : I \to C mono-filtered if every morphism A(i 1i 2)A(i_1 \to i_2) is a monomorphism in CC.

    =–

    +– {: .num_lemma #MonofilteredColimitsSendSheavesToSeparated}

    Lemma

    For CC a site and A:ISh(C)A : I \to Sh(C) a monofiltered diagram, the colimit over A:ISh(C)PSh(C)A : I \to Sh(C) \hookrightarrow PSh(C) is a separated presheaf.

    =–

    – {: .proof}

    Proof

    For {U αX}\{U_\alpha \to X\} any covering family in CC with S({U α})PSh(C)S(\{U_\alpha\}) \in PSh(C) the corresponding sieve, we need to show that

    lim iA i(X)Hom PSh(C)(S({U α}),lim iA i) \underset{\longrightarrow_i}{\lim} A_i(X) \to Hom_{PSh(C)}(S(\{U_\alpha\}), \underset{\longrightarrow_i}{\lim} A_i)

    is a monomorphism. An element on the left is represented by a pair (iI,aA i(X))(i \in I, a \in A_i(X)). Given any other such element, we may assume by filteredness that they are both represented over the same index ii. So let (i,a)(i,a) and (i,a)(i,a') be two such elements. They are different if they are different for all j>ij \gt i. Under the above function (i,a)(i,a) is mapped to the collection {i,a U α} α\{ i, a|_{U_\alpha} \}_\alpha and (i,a)(i,a') to {i,a U α} α\{ i, a'|_{U_\alpha} \}_\alpha. If aa is different from aa', then these families differ at stage ii. Then by mono-filteredness they differ also at all later stages, hence represent different families {U αlim iA i} α\{U_\alpha \to \underset{\to_i}{\lim} A_i\}_\alpha.

    =–

    +– {: .num_prop}

    Proposition

    For XX \in Mfd a compact manifold, the corresponding object XMfdSh(Mfd)X \in \mathrm{Mfd} \hookrightarrow \mathrm{Sh}(\mathrm{Mfd}) in the sheaf topos has the property that Hom Sh(Mfd)(X,)Hom_{Sh(Mfd)}(X,-) commutes with all mono-filtered colimits, def. \ref{MonoFilteredDiagram}.

    =–

    +– {: .proof}

    Proof

    Let A:ISh(Mfd)PSh(Mfd)A : I \to \mathrm{Sh}(\mathrm{Mfd}) \hookrightarrow \mathrm{PSh}(\mathrm{Mfd}) be a mono-filtered diagram of sheaves, which we regard as a diagram of presheaves. Write lim iA i\underset{\longrightarrow_i}\lim A_i for its colimit as such, hence as a diagram of presheaves. Since sheafification L:PSh(Mfd)Sh(Mfd)L : PSh(Mfd) \to Sh(Mfd) is left adjoined, the sheafification Llim iA iL \underset{\longrightarrow _i}{\lim} A_i of this colimit is the colimit of AA as a diagram of sheaves. By the Yoneda lemma and since colimits of presheaves are computed objectwise, it is sufficient to show that for XX a compact manifold, the value of the sheafified colimit is the colimit of the values of the sheaves on XX

    (Llim iA i)(X)(lim iA i)(X)lim iA i(X). (L \underset{\longrightarrow _i}{\lim} A_i)(X) \simeq (\underset{\longrightarrow _i}{\lim} A_i)(X) \simeq \underset{\longrightarrow _i}{\lim} A_i(X) \,.

    To see this, we evaluate the sheafification by the plus construction. By lemma \ref{MonofilteredColimitsSendSheavesToSeparated}, the presheaf lim iA i\underset{\longrightarrow_i}{\lim} A_i is already separated, so we obtain its sheafification by applying the plus-construction just once.

    We observe now that over a compact manifold XX the single plus-construction acts as the identity on the presheaf lim iA i\underset{\longrightarrow _i}{\lim} A_i.

    Namely the single +-construction over XX takes the colimit of the value of the presheaf on sieves

    S({U α}):=lim( α,βU α,β αU α) S(\{U_\alpha\}) := \underset{\longrightarrow}{\lim}( \coprod_{\alpha, \beta} U_{\alpha,\beta} \stackrel{\to}{\to} \coprod_\alpha U_\alpha)

    over all covers {U αX}\{U_\alpha \to X\} of XX. By the very definition of compactness, the inclusion of (the opposite category of) the category of finite covers of XX into that of all covers is a final functor. Therefore we may compute the +-construction over XX by the colimit over just the collection of finite covers. On a finite cover we have

    PSh(S({U α}),lim iA i) :=PSh(lim( α,βU αβ αU α),lim iA i) lim( αlim iA i(U α) α,βlim iA i(U α,β)) lim ilim( αA i(U α) α,βA i(U α,β)) lim iA i(X), \begin{aligned} \mathrm{PSh}(S(\{U_\alpha\}), \underset{\longrightarrow _i}{\lim} A_i) & := \mathrm{PSh}( \underset{\longrightarrow}{\lim}( \coprod_{\alpha,\beta} U_{\alpha \beta} \stackrel{\to}{\to} \coprod_\alpha U_\alpha), \underset{\longrightarrow _i}{\lim} A_i) \\ & \simeq \underset{\longleftarrow}{\lim} ( \prod_{\alpha } \underset{\longrightarrow_i}{\lim} A_i(U_{\alpha }) \stackrel{\to}{\to} \prod_{\alpha,\beta } \underset{\longrightarrow_i}{\lim} A_i(U_{\alpha,\beta}) ) \\ & \simeq \underset{\longrightarrow _i}{\lim}\underset{\longleftarrow}{\lim} ( \prod_{\alpha } A_i(U_{\alpha }) \stackrel{\to}{\to} \prod_{\alpha, \beta } A_i(U_{\alpha, \beta}) ) \\ & \simeq \underset{\longrightarrow _i}{\lim} A_i(X) \end{aligned} \,,

    where in the second but last step we used that filtered colimits commute with finite limits, and in the last step we used that each A iA_i is a sheaf.

    So in conclusion, for XX a compact manifold and A:ISh(Mfd)A : I \to Sh(Mfd) a monofiltered diagram, we have found that

    Hom Sh(Mfd)(X,Llim iA i) (lim iA i) +(X) lim iA i(X) lim iHom Sh(Mfd)(X,A i) \begin{aligned} Hom_{Sh(Mfd)}(X, L \underset{\longrightarrow_i}{\lim} A_i) & \simeq (\underset{\longrightarrow_i}{\lim} A_i)^+ (X) \\ & \simeq \underset{\longrightarrow_i}{\lim} A_i(X) \\ & \simeq \underset{\longrightarrow_i}{\lim} Hom_{Sh(Mfd)}(X, A_i) \end{aligned}

    =–

    • CommentRowNumber11.
    • CommentAuthorStephan A Spahn
    • CommentTimeApr 30th 2012
    • (edited Apr 30th 2012)

    Do you think this result can be improved from mono-filtered diagrams to filtered diagrams? The material on the plus construction is a bit sparse on the nlab; maybe we can extend it a bit…

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012
    • (edited Apr 30th 2012)

    Do you think this result can be improved from mono-filtered diagrams to filtered diagrams?

    Unfortunately I still can’t see it, no. But I am also still quite prepared not to be surprised if it turns out easy to see. I have forwarded the question to MO.

    The material on the plus construction is a bit sparse on the nlab; maybe we can extend it a bit…

    Yeah, I know. There is detailed discussion at sheaf that partly could go to plus construction. But in either case, these entries would deserve to be expanded considerably.

    Myself, I won’t find the time for it soon, though. Maybe somebody else feels like giving it a try?

  1. Myself, I won’t find the time for it soon, though. Maybe somebody else feels like giving it a try?

    Yes, I will try it with the description given in the Elephant.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    Great!

    • CommentRowNumber15.
    • CommentAuthorStephan A Spahn
    • CommentTimeApr 30th 2012
    • (edited Apr 30th 2012)

    I have given the two equivalent definitions (in terms a colimit over covering sieves and the one referring to equivalence classes of compatible families) from the Elephant and the one given by colimits over dense monomorphisms from sheaf.

    To address the question of compact objects in Sh (SmoothMfd)Sh_\infty(Smooth Mfd) there should be an (\infty,1)-plus construction, too. Is in this case where the (\infty,1)-site is just a 1-site somehow clear how this works?

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    Thanks! I have posted a Latest-Changes announcement here. Let’s further discuss there.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    As I just said there, for nn-stackification one will need to apply the plus-construction (n+2)(n+2)-times. Moreover, if the we are dealing with an \infty-presheaf AA with values in nn-types that is already separated, in that its descent morphism over a covering family {U iX}\{U_i \to X\}

    Hom(X,A)Hom(C({U i}),A) Hom(X, A) \to Hom( C(\{U_i\}), A )

    is, while not a homotopy equivalence, at least an iso on all positive homotopy groups, then a single further plus-construction would be sufficient to make it an nn-stack.

    This was my original strategy, for which the above proof is the easy version with n=0n = 0:

    1. first establish that the mono-filtered colimit of \infty-sheaves at the level of \infty-presheaves is a separated \infty-presheaf;

    2. then use that to \infty-stackify we need the plus-construction only once;

    3. then use that a single plus-construction evaluated on a compact XX acts trivially on that filtered \infty-colimit, because we may assume a finite, cover, which we may take inside the colimit, where it disappears to become an XX, since inside the \infty-colimit everything is \infty-sheaves;

    4. hence conclude that over a compact XX we may compute the \infty-colimit of \infty-sheaves by that of \infty-presheaves;

    5. and finally observe that therefore the above argument has shown that we may take XX inside the \infty-colimit.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2012
    • (edited May 1st 2012)

    A final thought before I have to go offline:

    the culprit is the ordinary notion of compact topological space as something admitting a finite subcover for any cover. Because this is a truncated notion.

    What we’d need instead in order to make the desired statement true is to consider spaces which admit a finite hypercover (of some height) refining any hypercover (of some height).

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2012

    Is that related to the notion of “strong compactness” i.e. of the geometric morphism Sh(X)SetSh(X)\to Set being “tidy” rather than merely “proper”? (Which is of course also only the 1-level version of a whole hierarchy of conditions of which ordinary compactness is the 0-level.)

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2012
    • (edited May 1st 2012)

    Something very closely related is true, I might just need to adjust my thoughts about it further:

    strongly compact toposes are characterized by having strongly compact sites and these are sites that are “compact sites” plus the additional requirement that, roughly, the intersections of a cover have an “effective cover”: in any case some extra condition on covers of the double intersections of a given cover, see around p. 59-60 in Moerdijk-Vermeulen.

    Moreover (this I am beginning to summarize here):

    • the sheaf topos over a compact and Hausdorff space is strongly compact;

    • the slice topos over a compact object is a strongly compact topos

      (while I don’t see this last item stated explicitly in Moerdijk-Vermeulen, this is immediate. For Stephan’s convenience I have spelled out a detailed proof here.

    • CommentRowNumber21.
    • CommentAuthorStephan A Spahn
    • CommentTimeMay 1st 2012
    • (edited May 1st 2012)
    • the sheaf topos over a compact and Hausdorff space is strongly compact;
    • the slice topos over a compact object is a strongly compact topos (while I don’t see this last item stated explicitly in Moerdijk-Vermeulen, this is immediate. For Stephan’s convenience I have spelled out a detailed proof here.

    Thanks, Urs. I am still reading Moerdijk-Vermeulen. Are in there statements of the converse type - i.e. if the petit or gros topos over XX is (strongly) compact then so is XX?

    Is that related to the notion of “strong compactness” i.e. of the geometric morphism Sh(X)SetSh(X)\to Set being “tidy” rather than merely “proper”? (Which is of course also only the 1-level version of a whole hierarchy of conditions of which ordinary compactness is the 0-level.)

    Shouldn’t we start counting at level (-1)?

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2012
    • (edited May 1st 2012)

    Are in there statements of the converse type

    Maybe not. I was thinking about that, too. Not sure yet.

    Shouldn’t we start counting at level (-1)?

    The note that I had made in the Definition section starts counting at (-1) to match with the truncation level of the objects involved in the definition.

    But these decisions are always a matter of taste at best. On the one hand it may seem bad to start counting at (-1), on the other it may be regarded as nicely indicating that the “origin of traditional thinking” which then is at 0 is indeed not the “absolute origin”. In either case, it’s just a convention that doesn’t matter too much.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2012
    • (edited May 1st 2012)

    Of course some converse statements are clear:

    Suppose a slice topos H /X\mathbf{H}_{/X} is strongly compact, hence forming sections Γ X():H /XSet\Gamma_X(-) : \mathbf{H}_{/X} \to Set preserves filtered colimits. Then in particular it preserves filtered colimits of trivial bundles of the form [X×F iX][X \times F_i \to X]. Here Γ X([X×F iX])=H(X,F i)\Gamma_X([X \times F_i \to X]) = \mathbf{H}(X, F_i) and hence also the converse statement holds:

    if H /X\mathbf{H}_{/X} is strongly compact then XHX \in \mathbf{H} is a compact object.

    I have expanded the proof here to display also this converse statement in detail.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2012
    • (edited May 1st 2012)

    Didn’t get to do much today (May day) and will have to go offline now. Just as a reminder for myself, here a sketch for how we should be able to prove that, after all, every compact smooth manifold XX is a compact object in Sh (SmthMfd)Sh_\infty(SmthMfd) and in particular in Sh(SmthMfd)Sh(SmthMfd).

    1. Present a given filtered \infty-diagram A :ISh (SmthMfd)A_\bullet : I \to Sh_\infty(SmthMfd) by a cofibrant-fibrant object A [I,[SmthMfd op,sSet] proj,loc] projA_\bullet \in [I, [SmthMfd^{op}, sSet]_{proj,loc}]_{proj}. Then in particular the ordinary colimit lim iA i\underset{\longrightarrow_i}{\lim} A_i presents the \infty-colimit in question.

    2. Observe that lim iA i\underset{\longrightarrow_i}{\lim} A_i is still fibrant in [SmthMfd op,sSet] proj[SmthMfd^{op}, sSet]_{proj} (because since the colimit is filtered, we can evaluate the lift against horns, which are finite simplicial sets, at some stage).

      Of course it will not be fibrant in [SmthMfd op,sSet] proj,loc[SmthMfd^{op}, sSet]_{proj, loc}. The \infty-colimit in question is presented by the fibrant replacement here (the \infty-stackification of the colimit).

    3. But since lim iA i\underset{\longrightarrow_i}{\lim} A_i is globally fibrant we can apply DHI, theorem 7.6 a) to represent a given morphism in Sh (SmthMfd)Sh_\infty(SmthMfd) from XX to the \infty-stackification as a morphism Ylim iA iY \to \underset{\longrightarrow_i}{\lim} A_i for YXY \to X a hypercover of XX.

    4. Now I think the differential geometric analog of DHI, prop 7.10 will hold: the hypercover over the smooth manifold XX (Hausdorff and all) can in fact be refined by a Čech cover.

    5. So then by compactness, that in turn can be refined by a finite Čech cover.

    6. And so we can evaluate the hom into the colimit out of the cover at some stage. Or more formally, homming the finite coend that gives the Čech nerve into the filtered colimit passes through the filtered colimit.

    I will try to make that into a proof tomorrow morning.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2012

    re point 4: I have put a proof into the nnLab, see here

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2012
    • (edited May 2nd 2012)

    I wrote:

    I will try to make that into a proof tomorrow morning.

    Here is a pdf with some notes. The proposition in question is the very last one. This is still meant to be experimental.

    That currently claims to show that for XMfdSh (Mfd)=:HX \in Mfd \hookrightarrow Sh_\infty(Mfd) =: \mathbf{H} a compact manifold, then H(X,)\mathbf{H}(X,-) preserves sequential \infty-colimits.

    It’s a bit technical and there may easily be mistakes in there. So handle with care.

    I guess the argument easily generalizes (if indeed correct, which you should try to doubt) from sequential to all filtered colimits, but I still need to check something for this.

    • CommentRowNumber27.
    • CommentAuthorStephan A Spahn
    • CommentTimeMay 4th 2012
    • (edited May 4th 2012)

    In

    pdf

    you refer (e.g. in Proposition 0.10) to ”Example ??”. This is used to argue that A jA jA_j\to A_j is a cofibration - but I guess this example lifts the statement in question from mono-filtered to sequential diagrams (in that case A iA jA_i\to A_j would be a monomorphism and we consider some projective model structure…)

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2012
    • (edited May 4th 2012)

    you refer (e.g. in Proposition 0.10) to ”Example ??”.

    Right, sorry, that pdf is taken out of the main file. This is now Example 2.3.15, page 130 here.

    The relation to the mono-filtered 1-colimits is a bit superficial, however, because here we are dealing now with homotopy colimits.

    But I am now fairly certain that I can complete the proof and generalize from sequential to all filtered colimits. We just need the kind of argument that I mention in the other thread and then the proof in that pdf goes through verbatim also for general filtered \infty-colimits.

    Right now I don’t have the leisure to write this up, though. But I am fairly confident now about the statement: for every compact manifold XX regarded under SmthMfdSh (SmthMfd):=HSmthMfd \hookrightarrow Sh_\infty(SmthMfd) := \mathbf{H} we have that H(X,)\mathbf{H}(X,-) commutes with filtered \infty-colimits of truncated objects.