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[… I had a question here which I have answered meanwhile – I’ll post something a little later …]
The subject looked so interesting!
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 $Sh(SmthMfd) \simeq Sh(CartSp)$.
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.
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!
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.
That looks very interesting. Have you any idea what the ’extra’ compact objects might be?
I have no idea how to characterize all of them. But if I am right that the closed $n$-disk is a compact object in $Sh(SmthMfd)$ then an obvious class of compact objects built from that are piecewise smooth finite CW-complexes. Things like this.
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???
True, orbifolds of compact manifolds should be compact objects in $Sh_\infty(SmthMfd)$.
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}
Call a filtered diagram $A : I \to C$ mono-filtered if every morphism $A(i_1 \to i_2)$ is a monomorphism in $C$.
=–
+– {: .num_lemma #MonofilteredColimitsSendSheavesToSeparated}
For $C$ a site and $A : I \to Sh(C)$ a monofiltered diagram, the colimit over $A : I \to Sh(C) \hookrightarrow PSh(C)$ is a separated presheaf.
=–
– {: .proof}
For $\{U_\alpha \to X\}$ any covering family in $C$ with $S(\{U_\alpha\}) \in PSh(C)$ the corresponding sieve, we need to show that
$\underset{\longrightarrow_i}{\lim} A_i(X) \to Hom_{PSh(C)}(S(\{U_\alpha\}), \underset{\longrightarrow_i}{\lim} A_i)$is a monomorphism. An element on the left is represented by a pair $(i \in I, a \in A_i(X))$. Given any other such element, we may assume by filteredness that they are both represented over the same index $i$. So let $(i,a)$ and $(i,a')$ be two such elements. They are different if they are different for all $j \gt i$. Under the above function $(i,a)$ is mapped to the collection $\{ i, a|_{U_\alpha} \}_\alpha$ and $(i,a')$ to $\{ i, a'|_{U_\alpha} \}_\alpha$. If $a$ is different from $a'$, then these families differ at stage $i$. Then by mono-filteredness they differ also at all later stages, hence represent different families $\{U_\alpha \to \underset{\to_i}{\lim} A_i\}_\alpha$.
=–
+– {: .num_prop}
For $X \in$ Mfd a compact manifold, the corresponding object $X \in \mathrm{Mfd} \hookrightarrow \mathrm{Sh}(\mathrm{Mfd})$ in the sheaf topos has the property that $Hom_{Sh(Mfd)}(X,-)$ commutes with all mono-filtered colimits, def. \ref{MonoFilteredDiagram}.
=–
+– {: .proof}
Let $A : I \to \mathrm{Sh}(\mathrm{Mfd}) \hookrightarrow \mathrm{PSh}(\mathrm{Mfd})$ be a mono-filtered diagram of sheaves, which we regard as a diagram of presheaves. Write $\underset{\longrightarrow_i}\lim A_i$ for its colimit as such, hence as a diagram of presheaves. Since sheafification $L : PSh(Mfd) \to Sh(Mfd)$ is left adjoined, the sheafification $L \underset{\longrightarrow _i}{\lim} A_i$ of this colimit is the colimit of $A$ as a diagram of sheaves. By the Yoneda lemma and since colimits of presheaves are computed objectwise, it is sufficient to show that for $X$ a compact manifold, the value of the sheafified colimit is the colimit of the values of the sheaves on $X$
$(L \underset{\longrightarrow _i}{\lim} A_i)(X) \simeq (\underset{\longrightarrow _i}{\lim} A_i)(X) \simeq \underset{\longrightarrow _i}{\lim} A_i(X) \,.$To see this, we evaluate the sheafification by the plus construction. By lemma \ref{MonofilteredColimitsSendSheavesToSeparated}, the presheaf $\underset{\longrightarrow_i}{\lim} A_i$ is already separated, so we obtain its sheafification by applying the plus-construction just once.
We observe now that over a compact manifold $X$ the single plus-construction acts as the identity on the presheaf $\underset{\longrightarrow _i}{\lim} A_i$.
Namely the single +-construction over $X$ takes the colimit of the value of the presheaf on sieves
$S(\{U_\alpha\}) := \underset{\longrightarrow}{\lim}( \coprod_{\alpha, \beta} U_{\alpha,\beta} \stackrel{\to}{\to} \coprod_\alpha U_\alpha)$over all covers $\{U_\alpha \to X\}$ of $X$. By the very definition of compactness, the inclusion of (the opposite category of) the category of finite covers of $X$ into that of all covers is a final functor. Therefore we may compute the +-construction over $X$ by the colimit over just the collection of finite covers. On a finite cover we have
$\begin{aligned} \mathrm{PSh}(S(\{U_\alpha\}), \underset{\longrightarrow _i}{\lim} A_i) & := \mathrm{PSh}( \underset{\longrightarrow}{\lim}( \coprod_{\alpha,\beta} U_{\alpha \beta} \stackrel{\to}{\to} \coprod_\alpha U_\alpha), \underset{\longrightarrow _i}{\lim} A_i) \\ & \simeq \underset{\longleftarrow}{\lim} ( \prod_{\alpha } \underset{\longrightarrow_i}{\lim} A_i(U_{\alpha }) \stackrel{\to}{\to} \prod_{\alpha,\beta } \underset{\longrightarrow_i}{\lim} A_i(U_{\alpha,\beta}) ) \\ & \simeq \underset{\longrightarrow _i}{\lim}\underset{\longleftarrow}{\lim} ( \prod_{\alpha } A_i(U_{\alpha }) \stackrel{\to}{\to} \prod_{\alpha, \beta } A_i(U_{\alpha, \beta}) ) \\ & \simeq \underset{\longrightarrow _i}{\lim} A_i(X) \end{aligned} \,,$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 $A_i$ is a sheaf.
So in conclusion, for $X$ a compact manifold and $A : I \to Sh(Mfd)$ a monofiltered diagram, we have found that
$\begin{aligned} Hom_{Sh(Mfd)}(X, L \underset{\longrightarrow_i}{\lim} A_i) & \simeq (\underset{\longrightarrow_i}{\lim} A_i)^+ (X) \\ & \simeq \underset{\longrightarrow_i}{\lim} A_i(X) \\ & \simeq \underset{\longrightarrow_i}{\lim} Hom_{Sh(Mfd)}(X, A_i) \end{aligned}$=–
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…
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?
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.
Great!
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 $Sh_\infty(Smooth Mfd)$ 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?
Thanks! I have posted a Latest-Changes announcement here. Let’s further discuss there.
As I just said there, for $n$-stackification one will need to apply the plus-construction $(n+2)$-times. Moreover, if the we are dealing with an $\infty$-presheaf $A$ with values in $n$-types that is already separated, in that its descent morphism over a covering family $\{U_i \to X\}$
$Hom(X, A) \to Hom( C(\{U_i\}), A )$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 $n$-stack.
This was my original strategy, for which the above proof is the easy version with $n = 0$:
first establish that the mono-filtered colimit of $\infty$-sheaves at the level of $\infty$-presheaves is a separated $\infty$-presheaf;
then use that to $\infty$-stackify we need the plus-construction only once;
then use that a single plus-construction evaluated on a compact $X$ 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 $X$, since inside the $\infty$-colimit everything is $\infty$-sheaves;
hence conclude that over a compact $X$ we may compute the $\infty$-colimit of $\infty$-sheaves by that of $\infty$-presheaves;
and finally observe that therefore the above argument has shown that we may take $X$ inside the $\infty$-colimit.
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).
Is that related to the notion of “strong compactness” i.e. of the geometric morphism $Sh(X)\to Set$ 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.)
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.
- 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 $X$ is (strongly) compact then so is $X$?
Is that related to the notion of “strong compactness” i.e. of the geometric morphism $Sh(X)\to Set$ 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)?
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.
Of course some converse statements are clear:
Suppose a slice topos $\mathbf{H}_{/X}$ is strongly compact, hence forming sections $\Gamma_X(-) : \mathbf{H}_{/X} \to Set$ preserves filtered colimits. Then in particular it preserves filtered colimits of trivial bundles of the form $[X \times F_i \to X]$. Here $\Gamma_X([X \times F_i \to X]) = \mathbf{H}(X, F_i)$ and hence also the converse statement holds:
if $\mathbf{H}_{/X}$ is strongly compact then $X \in \mathbf{H}$ is a compact object.
I have expanded the proof here to display also this converse statement in detail.
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 $X$ is a compact object in $Sh_\infty(SmthMfd)$ and in particular in $Sh(SmthMfd)$.
Present a given filtered $\infty$-diagram $A_\bullet : I \to Sh_\infty(SmthMfd)$ by a cofibrant-fibrant object $A_\bullet \in [I, [SmthMfd^{op}, sSet]_{proj,loc}]_{proj}$. Then in particular the ordinary colimit $\underset{\longrightarrow_i}{\lim} A_i$ presents the $\infty$-colimit in question.
Observe that $\underset{\longrightarrow_i}{\lim} A_i$ is still fibrant in $[SmthMfd^{op}, sSet]_{proj}$ (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 $[SmthMfd^{op}, sSet]_{proj, loc}$. The $\infty$-colimit in question is presented by the fibrant replacement here (the $\infty$-stackification of the colimit).
But since $\underset{\longrightarrow_i}{\lim} A_i$ is globally fibrant we can apply DHI, theorem 7.6 a) to represent a given morphism in $Sh_\infty(SmthMfd)$ from $X$ to the $\infty$-stackification as a morphism $Y \to \underset{\longrightarrow_i}{\lim} A_i$ for $Y \to X$ a hypercover of $X$.
Now I think the differential geometric analog of DHI, prop 7.10 will hold: the hypercover over the smooth manifold $X$ (Hausdorff and all) can in fact be refined by a Čech cover.
So then by compactness, that in turn can be refined by a finite Čech cover.
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.
re point 4: I have put a proof into the $n$Lab, see here
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 $X \in Mfd \hookrightarrow Sh_\infty(Mfd) =: \mathbf{H}$ a compact manifold, then $\mathbf{H}(X,-)$ 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.
In
you refer (e.g. in Proposition 0.10) to ”Example ??”. This is used to argue that $A_j\to A_j$ is a cofibration - but I guess this example lifts the statement in question from mono-filtered to sequential diagrams (in that case $A_i\to A_j$ would be a monomorphism and we consider some projective model structure…)
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 $X$ regarded under $SmthMfd \hookrightarrow Sh_\infty(SmthMfd) := \mathbf{H}$ we have that $\mathbf{H}(X,-)$ commutes with filtered $\infty$-colimits of truncated objects.
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