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I am not too convinced that the comparison lemma at dense sub-site is correct. Here is my proposed counterexample:
Consider the following category $\mathcal{C}$, where everything commute:
$\array{ D & \rightarrow & C\\ \downarrow & ^\mathllap{e}\swarrow\nearrow_{\mathrlap{f}}& \downarrow\\ B & \rightarrow & A }$In particular, $e$ and $f$ are isomorphisms.
We consider the topology $J$ where $J(B), J(C), and J(D)$ all consist only of the maximal sieve, and $J(A)$ consists of the maximal sieve and (maximal sieve$\setminus id_A$). This can be manually checked to be a Grothendieck topology.
Consider the subcategory $\mathcal{D}$ of $\mathcal{C}$ consisting of the same objects and arrows except without the arrows $e$ and $f$. Then $\mathcal{D}$ is dense according to the definition ($e$ can be covered by $f$, and vice versa).
$\array{ D & \rightarrow & C\\ \downarrow & & \downarrow\\ B & \rightarrow & A }$Then the restriction of a sheaf on $\mathcal{C}$ to $\mathcal{D}$ need not be a sheaf. Indeed, consider the sheaf that maps $\mathcal{C}$ to
$\array{ 1 & \overset{!}\leftarrow & X\\ \mathllap{!}\uparrow & //& ||\\ X & = & X }$where $X$ is any set with at least two elements (the $//$ is supposed to be a diagonal equals sign). Then the restriction to $\mathcal{D}$ gives
$\array{ 1 & \overset{!}\leftarrow & X\\ \mathllap{!}\uparrow & & ||\\ X & = & X }$which is not a sheaf, since if $a \neq b \in X$, then $a \in F(B)$, $b \in F(C)$ plus the unique element in $F(D) = 1$ is a matching family on the cover that includes everything but the identity on $A$, but this does not come from any element of $F(A)$. So the image of a sheaf under the restriction map need not be a sheaf.
The comparison lemma originates with Verdier’s exposé III pp265ff in SGA 4, but the more general form seems to stem from Kock and Moerdijk here. If one translates their version on p.151f to your situation it seems that their condition 2) is violated by your example:
For your morphism $f:B\to C$ in $\mathcal{C}$ there must be a covering sieve $\{\xi_i\colon X_i\to B\}$ in $\mathcal{D}$ (!) such that for all indices $f\circ \xi_i$ is in $\mathcal{D}$ but since the only cover on $B$ is the maximal sieve $f\circ id_B$ fails this.
So my guess here is that the second condition in def.2.2 at dense sub-site has to be read as saying that the covering family $\{f_i\colon U_i\to U\}$ is in $\mathcal{D}$. (So the nLab is probably correct, but I have to run !?)
According to the nLab version, using your notation, the source $B$ need not be an element of $\mathcal{D}$, so it doesn’t make sense to require that the covering sieve is (generated by maps) in $\mathcal{D}$.
I would be inclined to believe that the Kock and Moerdijk version is correct, where we only talk about maps $f: B \to C$ with $B, C \in \mathcal{D}$, and then require that the covering family lies in $\mathcal{D}$.
Yes, you are right, the version on the nLab has a mistake/typo: the domain of $f$ and the covering family must also be in $D$. Someone should fix it. (-: (If they felt especially energetic, they might even mention the generalization to the case of an arbitrary functor between sites, not necessarily a subcategory inclusion.)
If they felt especially energetic, they might even mention the generalization to the case of an arbitrary functor between sites, not necessarily a subcategory inclusion
Wouldn’t that belong more to the comparison lemma page, since that would no longer be a subsite? Or rather, should the two pages be merged? After all, a dense subsite is just a subcategory for which the comparison lemma holds.
Yikes, I didn’t realize we had both of those. At the moment they are totally redundant with each other; the comparison lemma page starts with the even less general version for full subcategories, then cites a 2011 Beilinson paper for the faithful functor version, apparently not realizing that Kock and Moerdijk proved an even more general version 20 years earlier. I would be inclined to merge “dense subsite” into “comparison lemma” (and, maybe, call it something like “dense morphism of sites”; it doesn’t really fit our conventions to have a page named after a lemma). Other opinions?
The version in the comparison lemma page has a slightly different hypothesis. I think we could present “comparison lemma/dense subsite” as a general concept of functors between sites inducing equivalences of sheaf categories, and then present the different versions.
I am not sure that there is anything wrong with the definition at dense sub-site: though a bit counter-intuitive it might be just an elephantine extra-generality added to the Kock-Moerdijk version. Note that the example in #1 is ruled out by it since again $f\circ id_B$ is not in $\mathcal{D}$.
Secondly, I am not very happy with this merging business unless the server runs out of storage. In my view it’s perfectly legitimate to have a seperate page for the definition of dense sub-site (a notion that has the elephant’s blessing and e.g. avoids the trouble to unwind the Kock-Moerdijk definition in a case of a mere subcategory inclusion!) and then it is a politeness to the reader to give additional information on the comparison lemma there even if a separate page on this exists elsewhere on the nLab in this case. Some redundancy is essential for communication!
So I would think it best to expand dense sub-site with a remark on the more general version and put the different versions of site morphisms at comparison lemma. Then one still can refer to dense sub-site from elsewhere without tracking the reader through all the details of the comparison lemma.
In any case, I have thrown in some references, until the smoke over the definition settles.
For the map $f: B \to C$, the source $B$ is covered by $e$, and the composite $f \circ e = \id_C$ is in $\mathcal{D}$. So this should satisfy the definition.
I agree that we could maintain two separate pages, and the version for subsites would be convenient, but I think some restructuring would be needed to make it clear how the ideas relate.
I read
- for every morphism $f : U \to d$ in $C$ with $d \in D$ there is a covering family $\{f_i : U_i \to U\}$ such that the composites $f \circ f_i$ are in $D$.
as saying that $\{f_i : U_i \to U\}$ is a cover (i.e. a sieve in $J(U)$) on $U$ in $\mathcal{C}$, at least that’s what reading the elephant suggests as well as the analogy with Kock-Moerdijk.
The elephant says “a sieve generated by morphisms satsifying this property”. Also $\{f_i: U_i \to U\}$ couldn’t be a sieve in $\mathcal{C}$ unless there are no maps from elements of $\mathcal{C} \setminus \mathcal{D}$ to the $U_i$ (and it couldn’t be a sieve in $\mathcal{D}$ if $U$ is not an element of $\mathcal{D}$).
It seems that we agree on what the correct theorem should be, and we just disagree on whether the nLab phrasing of it is correct. I’ve updated the definition at dense sub-site to make it clearer.
Maybe I am confused here but doesn’t the elephant right behind def.2.2.1 stress the point that in $f:U\to V$ the source $U$ does not have to be in $\mathcal{D}$ ?
That’s what the Elephant says, and I believe #1 is a counterexample to the Elephant version (but not the current nLab version and the Kock-Moerdijk version).
It would perhaps suffice to assume the maps $g:W\to U$ in the elephant to be in $\mathcal{D}$ meaning that the old nLab version might be viable after all if one specifies that the cover $J(U)$ is generated by maps in $\mathcal{D}$ thereby ruling out the example. [well, this obviously implies that U is $\mathcal{D}$ and we end up with the current nLab version again! hmm..]
In any case this still seems highly involved in comparison with the more intuitive version à la Kock-Moerdijk and I am not sure how much mileage can be drawn from this generality in the end since the elephant probably just uses it to prove the Kock-Moerdijk result on étendues on p.769.
Regarding the page merging, I was not suggesting that the merged page include only the more general version, or even necessarily start with it. I am not sure what the advantage would be of having two pages with essentially the same content.
I think I agree that the version in the Elephant is wrong. Here is a simpler counterexample: let $C$ be any groupoid, with the trivial topology (only maximal sieves cover), and let $D$ be the discrete category on the same objects. Then for any morphism $f:U\to V$, its inverse $f^{-1}:V\to U$ generates the maximal sieve on $U$, and the composite $f f^{-1} = 1_V$ is in $D$, so the Elephant’s definition is satisfied. But the restriction $Set^{C^{op}} \to Set^{D^{op}}$ is not generally an equivalence.
@#15. My point, is that when one merges the pages one is more likely to end up with the infinimum of readibility or even below, unless one is willing to invest an unreasonable amount of work that is better allocated to other tasks. In an ideal (and I would think maybe impossible) n-world one has two pages one on the dense sub-site because that is an notion that people might want to look up which points a bit to the ramifications and a nice page on the comparison lemma in all its glory&proofs so the ultimate difference would be in emphasis and elaboration.
Basically the idea is to let float the comparison lemma as an ugly duckling for the time being and hope that it will eventually develop into a beautiful swan (presumably by the hands of the mythic energetic person) instead of merging the child with the bathtub. I don’t think that the duckling does any harm any way.
In my view it is better to relax and learn to live with the enormouous mess that the nLab is, and do improvements on less principled and more casual grounds, after all, since the hard core of regular contributors is just a half a dozen people - one should stay realistic about what can be done. In the end merging is a highly intrusive way of editing and should like other editing done in respect to the intentions and work invested by the previous contributors.
I have added Mike’s example and a remark on the old definition at dense sub-site.
Here is something in dense site that I don’t understand:
For C=TopManifold the category of all paracompact topological manifolds equipped with the open cover coverage, the category CartSp_top is a dense sub-site: every paracompact topological manifold has a good open cover by open balls homeomorphic to a Cartesian space.
As far as I know, even for the E_8 topological 4-manifold this is an open problem: we don’t know whether the E_8-manifold admits a cover whose finite intersections are homeomorphic (and not merely homotopy equivalent) to open balls.
The page dense site doesn’t exist. The statement in question was given by Urs in Rev #1 of dense sub-site.
Re #20: thanks for the correction. Should I correct the (apparent) mistake?
This might be a stupid question, but why do you need the intersections to be homeomorphic to open balls?
So that the category consisting just of open balls is a dense subsite, rather than the larger category of spaces homotopy equivalent to open balls, which consists of all contractible topological manifolds. This is presumably a bit more unwieldy, but formally I guess it’s ok.
Dmitri, is the assertion in the nLab at least conjectured to be true? Is dimension 4 the only dimension where it is not known to be true?
It might be the issue of non-triangulable manifolds in many dimensions, after the work of Manolescu (spelling?)
Triangulations give rise to good covers (take the stars of all vertices). Constructing a triangulation from a good cover seems to be more difficult, but perhaps it is not unreasonable to assume it can be done.
Thus I would guess that nontriangulable topological manifolds do not admit good open covers. The work of Manolescu shows that nontriangulable manifolds exist in dimensions 5 and higher.
I might still be missing something obvious - doesn’t the definition of being a dense subsite just require the existence of an open cover by open balls, without regards to what the intersections look like?
Re #27: I think this is correct, one doesn’t need the existence of good covers to show that sheaves of sets on manifolds and cartesian manifolds are the same, a cover by cartesian manifolds (with no restrictions on intersections) seems to be sufficient.
In fact, I don’t think this is necessary for simplicial presheaves either.
But probably a lot of nice properties don’t work, no? We don’t get Čech nerves being levelwise a coproduct of representables.
Re #29: For topological manifolds one has good hypercovers, which suffice for all purposes.
For now I deleted the word “good” before “open cover” in the dense sub-site article. This corrects the mistake, at least.
We wouldn’t need the manifolds to be paracompact if we use open covers. I’ve split it into two examples manifold/open cover and paracompact manifold/good open cover. Someone else might be able to come up with a better name to distinguish the categories of topological manifolds and paracompact topological manifolds.
Re #32: I still don’t see a reason why paracompact topological manifolds should admit a good open cover.
In fact, https://ncatlab.org/nlab/show/good+open+cover#nonexistence_for_topological_manifolds almost explicitly claims they don’t exist in general.
Dmitri, you are right. Thanks for catching this. I remember that I had fixed this at some point, but apparently not in all entries. For instance here at Euclidean-topological infinity-groupoid I had added the explicit extra condition “such that a good open cover exists”. But of course you are right that for the intended conclusion one does not even need this assumption.
Hum. I guess I was worried about hypercompleteness. But never mind.
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