Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorCharles Rezk
   • CommentTimeJun 28th 2016
   • PermaLink
   Author: Charles Rezk
   Format: MarkdownItexI don't understand why, as stated in [Numerable open covers](https://ncatlab.org/nlab/show/numerable+open+cover), numerable covers are supposed to form a Grothendieck site. Where is this proved?

   I don’t understand why, as stated in Numerable open covers, numerable covers are supposed to form a Grothendieck site. Where is this proved?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 28th 2016
   • (edited Jun 28th 2016)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThat entry seems to have arisen from the discussion [here](https://nforum.ncatlab.org/discussion/1243/definition-at-numerable-open-cover/) and [here](https://nforum.ncatlab.org/discussion/1386/pretopologies-on-top/).

   That entry seems to have arisen from the discussion here and here.

3. • CommentRowNumber3.
   • CommentAuthorCharles Rezk
   • CommentTimeJun 28th 2016
   • PermaLink
   Author: Charles Rezk
   Format: MarkdownItexOkay, I'm guessing the definition of numerable covering given on the nLab page is just wrong. Following the linked article by Dold, "Partitions of unity in the theory of fibrations", we see that introduction he writes in the introduction (speaking of certain properties $P$ of continuous maps with target $B$): * Roughly speaking, we show that each of these properties $P$ is a local property with respect to $B$. More precisely, $P$ holds provided it holds over every set of a numerable covering $\{V_\lambda\}_{\lambda\in \Lambda}$ of $B$. *Numerable* means there exists a locally finite partition of unity $\pi_\gamma\colon B\to [0,1]$, $\gamma\in \Gamma$, such that the covering $\{\pi_\gamma^{-1}(0,1]\}$ refines $\{V_\lambda\}$. Note that Dold only requires $\pi^{-1}_\gamma(0,1]$ to be contained in some $V_\lambda$, not its closure (as in the nLab page). Also, I suspect "locally finite partition of unity" means not merely *pointwise finite* (as in the nLab page), but rather that every point $b\in B$ has a neighborhood that intersects only finitely many $\pi_\gamma^{-1}(0,1]$. This is supported by May's definition of numerable cover (in "Consise Introduction", section 7.4), which is similar to (though maybe not identical to) Dold's definition. With this definition, it looks a lot more plausible to me that these form a site. In particular, it's clear that partial sums $\sum_{\gamma\in S}\pi_\gamma$ for subsets $S\subseteq \Gamma$ always define continuous functions $B\to [0,1]$, which you would seem to need to assemble numerable covers $\{U_{\lambda\delta}\to V_\lambda\}$ of a numerable cover $\{V_\lambda\to B\}$ into a numerable cover of $B$.

   Okay, I’m guessing the definition of numerable covering given on the nLab page is just wrong. Following the linked article by Dold, “Partitions of unity in the theory of fibrations”, we see that introduction he writes in the introduction (speaking of certain properties $P$ of continuous maps with target $B$):

   • Roughly speaking, we show that each of these properties $P$ is a local property with respect to $B$. More precisely, $P$ holds provided it holds over every set of a numerable covering $\{V_\lambda\}_{\lambda\in \Lambda}$ of $B$. Numerable means there exists a locally finite partition of unity $\pi_\gamma\colon B\to [0,1]$, $\gamma\in \Gamma$, such that the covering $\{\pi_\gamma^{-1}(0,1]\}$ refines $\{V_\lambda\}$.

   Note that Dold only requires $\pi^{-1}_\gamma(0,1]$ to be contained in some $V_\lambda$, not its closure (as in the nLab page). Also, I suspect “locally finite partition of unity” means not merely pointwise finite (as in the nLab page), but rather that every point $b\in B$ has a neighborhood that intersects only finitely many $\pi_\gamma^{-1}(0,1]$. This is supported by May’s definition of numerable cover (in “Consise Introduction”, section 7.4), which is similar to (though maybe not identical to) Dold’s definition.

   With this definition, it looks a lot more plausible to me that these form a site. In particular, it’s clear that partial sums $\sum_{\gamma\in S}\pi_\gamma$ for subsets $S\subseteq \Gamma$ always define continuous functions $B\to [0,1]$, which you would seem to need to assemble numerable covers $\{U_{\lambda\delta}\to V_\lambda\}$ of a numerable cover $\{V_\lambda\to B\}$ into a numerable cover of $B$.

4. • CommentRowNumber4.
   • CommentAuthorCharles Rezk
   • CommentTimeJun 28th 2016
   • PermaLink
   Author: Charles Rezk
   Format: MarkdownItexProbably to have a site, I also want to keep the condition that closures of $\pi_\gamma^{-1}(0,1]$ are in some $V_\lambda$.

   Probably to have a site, I also want to keep the condition that closures of $\pi_\gamma^{-1}(0,1]$ are in some $V_\lambda$.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2016
   • (edited Jun 28th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe two edits to the entry that are not purely formatting are [version 1](https://ncatlab.org/nlab/revision/numerable+open+cover/1) and [revision 3](https://ncatlab.org/nlab/revision/diff/numerable+open+cover/3), both by David Roberts from 7 years ago. I suppose when David comes back online in a few hours he will see this here and react.

   The two edits to the entry that are not purely formatting are version 1 and revision 3, both by David Roberts from 7 years ago. I suppose when David comes back online in a few hours he will see this here and react.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 28th 2016
   • (edited Jun 29th 2016)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexWow, why do we only have pointwise finite? That's never the definition, though one can adjust the collection of functions if it is merely pointwise finite to be locally finite: this is in Dold's _Lectures on algebraic topology_, for instance. If one merely asks for the _existence_ of a partition of unity subordinate to an open cover in Dold's sense, one can find a partition of unity with slightly smaller supports that still induces an open cover, no? As to where this is proved, I (believed I) proved it, but have never published it. Even if numerable covers do not form a pretopology, I'm convinced they at least form a [[coverage]], hence a site. In particular, one doesn't actually need the comment in Charles' last paragraph (assembling together numerable covers) to get a coverage. Edit: I've seen the stronger 'support contained inside the open' definition around the place, and I'm happy to keep that: do we just need to make a remark that this is slightly different to Dold's version? Or are there results that hinge on his exact condition?

   Wow, why do we only have pointwise finite? That’s never the definition, though one can adjust the collection of functions if it is merely pointwise finite to be locally finite: this is in Dold’s Lectures on algebraic topology, for instance.

   If one merely asks for the existence of a partition of unity subordinate to an open cover in Dold’s sense, one can find a partition of unity with slightly smaller supports that still induces an open cover, no?

   As to where this is proved, I (believed I) proved it, but have never published it. Even if numerable covers do not form a pretopology, I’m convinced they at least form a coverage, hence a site. In particular, one doesn’t actually need the comment in Charles’ last paragraph (assembling together numerable covers) to get a coverage.

   Edit: I’ve seen the stronger ’support contained inside the open’ definition around the place, and I’m happy to keep that: do we just need to make a remark that this is slightly different to Dold’s version? Or are there results that hinge on his exact condition?

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 29th 2016
   • (edited Jun 29th 2016)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexOk, Dold's book that I mmentioned in #6 , in A.2.14 shows that numerable covers in his sense form a coverage, hence a site. I can edit this in to the entry. I don't claim my observing this fact to be incredibly ground-breaking, I've been saying for a long time it should have been obvious that classical algebraic topology is on the numerable site a lot if the time, hence all the restriction to paracompact spaces where the open covers site and the numerable site agree.

   Ok, Dold’s book that I mmentioned in #6 , in A.2.14 shows that numerable covers in his sense form a coverage, hence a site. I can edit this in to the entry. I don’t claim my observing this fact to be incredibly ground-breaking, I’ve been saying for a long time it should have been obvious that classical algebraic topology is on the numerable site a lot if the time, hence all the restriction to paracompact spaces where the open covers site and the numerable site agree.

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 29th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI've added some clarifying remarks around the claim there is a site (NB it didn't say _Grothendieck_ site, which would imply a Grothendieck (pre)topology, when all I claim is a coverage). I think we need to sort out which definition we adopt, in particular noting both conventions and which gives stability under composing covers)

   I’ve added some clarifying remarks around the claim there is a site (NB it didn’t say Grothendieck site, which would imply a Grothendieck (pre)topology, when all I claim is a coverage). I think we need to sort out which definition we adopt, in particular noting both conventions and which gives stability under composing covers)

9. • CommentRowNumber9.
   • CommentAuthorCharles Rezk
   • CommentTimeJun 30th 2016
   • (edited Jun 30th 2016)
   • PermaLink
   Author: Charles Rezk
   Format: MarkdownItexThanks, David, that helps a lot. BTW, I don't actually *need* the numerable site ... but I've been trying to puzzle out a bunch of things related to equivariant bundles and classifying spaces, and those sorts of theorems often involve a numerability hypothesis.

   Thanks, David, that helps a lot. BTW, I don’t actually need the numerable site … but I’ve been trying to puzzle out a bunch of things related to equivariant bundles and classifying spaces, and those sorts of theorems often involve a numerability hypothesis.