Hum. I guess I was worried about hypercompleteness. But never mind.

]]>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.

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.

]]>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.

]]>For now I deleted the word “good” before “open cover” in the dense sub-site article. This corrects the mistake, at least.

]]>Re #29: For topological manifolds one has good hypercovers, which suffice for all purposes.

]]>But probably a lot of nice properties don’t work, no? We don’t get Čech nerves being levelwise a coproduct of representables.

]]>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.

]]>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?

]]>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.

]]>It might be the issue of non-triangulable manifolds in many dimensions, after the work of Manolescu (spelling?)

]]>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?

]]>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.

]]>This might be a stupid question, but why do you need the intersections to be homeomorphic to open balls?

]]>Re #20: thanks for the correction. Should I correct the (apparent) mistake?

]]>The page dense site doesn’t exist. The statement in question was given by Urs in Rev #1 of 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.

]]>I have added Mike’s example and a remark on the old definition at dense sub-site.

]]>@#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 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.

]]>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.

]]>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.

]]>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).

]]>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 $\mathcal{D}$ ?