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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 31st 2010
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   Author: DavidRoberts
   Format: MarkdownItexRegulars here would know that I harp on about the numerable site (i.e. Top with numerable open covers as covers). One reflection of the invisible ubiquity of numerable covers is that Milnor's universal $G$-bundle is only universal for $G$-bundles that trivialise over numerable covers, as it is itself numerable, and the 'standard' trivialising cover pulls back along a classifying map to give a trivialising cover. I have said that focusing on pretopologies on Top other than that of open covers is analogous to looking at the various pretopologies on the category of schemes: some 'work' for nice schemes, some 'work' for nasty schemes and so on. Except in this case instead of moving to covers that are more general (e.g. etale or Nisnevich) than the natural ones (Zariski or open immersions), we are moving to more specific covers, namely open covers induced by pulling back along certain coprobes (viz. partitions of unity). One contrast to the category of schemes is that there 'aren't enough' Zariski covers, and so we want more covers, but there are perhaps not enough numerable covers, and so one could question whether it is sensible to go from a situation where there are certainly plenty of covers (all open covers of a topological space) to a more restricted notion. Perhaps the guiding notion is wanting spaces to be locally trivial: a manifold is locally contractible, for example. But then one has to question what 'locally trivial' means: the Warsaw circle has trivial fundamental groups, but has non-trivial first shape homotopy group. There are spaces which have local pathologies so bad that no non-trivial connected covering spaces exist, and even with weaker notions of covering space there are not enough of these. There is of course the option of passing to more general covers, like when dealing with Lie groupoids, people pass to looking at surjective local diffeomorphisms or surjective submersions instead of open covers, but these are essentially the same as open covers (they generate the same Grothendieck topology). This is of course possible for topological spaces (local homeomorphisms, topological submersions = admits local sections through every point for example), but unlike numerable covers, there don't seem to be many fine distinctions between which covers are good and which aren't. One pretopology which is very much 'bigger' than open covers is the class of open surjections. These are 'good' in the sense that they are effective descent morphisms in Top. This is certainly not equivalent to open covers for some class of spaces. For example the path fibration is an open surjection for locally path connected spaces, but does not admit local sections over an open cover if the space is not semi-locally contractible. I do not know if there is a category of spaces (say CW complexes at least, or triangulable spaces; perhaps restrict to cellular maps) such that any open surjection admits local sections over an open cover. One thing which people might suggest is that the whole lot would be better encoded on passing to some sort of topos of sheaves, but of course the choice of (pre)topology is important. So perhaps at best we could work with a category of presheaves, but this is sort of getting away from spaces as we know them (in the most general sense, even).

   Regulars here would know that I harp on about the numerable site (i.e. Top with numerable open covers as covers). One reflection of the invisible ubiquity of numerable covers is that Milnor’s universal $G$-bundle is only universal for $G$-bundles that trivialise over numerable covers, as it is itself numerable, and the ’standard’ trivialising cover pulls back along a classifying map to give a trivialising cover. I have said that focusing on pretopologies on Top other than that of open covers is analogous to looking at the various pretopologies on the category of schemes: some ’work’ for nice schemes, some ’work’ for nasty schemes and so on. Except in this case instead of moving to covers that are more general (e.g. etale or Nisnevich) than the natural ones (Zariski or open immersions), we are moving to more specific covers, namely open covers induced by pulling back along certain coprobes (viz. partitions of unity).

   One contrast to the category of schemes is that there ’aren’t enough’ Zariski covers, and so we want more covers, but there are perhaps not enough numerable covers, and so one could question whether it is sensible to go from a situation where there are certainly plenty of covers (all open covers of a topological space) to a more restricted notion. Perhaps the guiding notion is wanting spaces to be locally trivial: a manifold is locally contractible, for example. But then one has to question what ’locally trivial’ means: the Warsaw circle has trivial fundamental groups, but has non-trivial first shape homotopy group. There are spaces which have local pathologies so bad that no non-trivial connected covering spaces exist, and even with weaker notions of covering space there are not enough of these.

   There is of course the option of passing to more general covers, like when dealing with Lie groupoids, people pass to looking at surjective local diffeomorphisms or surjective submersions instead of open covers, but these are essentially the same as open covers (they generate the same Grothendieck topology). This is of course possible for topological spaces (local homeomorphisms, topological submersions = admits local sections through every point for example), but unlike numerable covers, there don’t seem to be many fine distinctions between which covers are good and which aren’t.

   One pretopology which is very much ’bigger’ than open covers is the class of open surjections. These are ’good’ in the sense that they are effective descent morphisms in Top. This is certainly not equivalent to open covers for some class of spaces. For example the path fibration is an open surjection for locally path connected spaces, but does not admit local sections over an open cover if the space is not semi-locally contractible. I do not know if there is a category of spaces (say CW complexes at least, or triangulable spaces; perhaps restrict to cellular maps) such that any open surjection admits local sections over an open cover.

   One thing which people might suggest is that the whole lot would be better encoded on passing to some sort of topos of sheaves, but of course the choice of (pre)topology is important. So perhaps at best we could work with a category of presheaves, but this is sort of getting away from spaces as we know them (in the most general sense, even).

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 31st 2010
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   Author: DavidRoberts
   Format: MarkdownItexSomething that just occurred to me is that usual open covers are given by things which are pullbacks of the open point in the [[Sierpinski space]] $\Sigma = \{0,1\}$ (this being a sort of strong subobject classifier - speaking heuristically), whereas partitions (not of unity, but of some arbitrary function $\sum_i u_i:X \to \mathbb{R}^+$) $\{u_i:X \to \mathbb{R}_{\ge 0}\}$ give rise to a cover by pulling back the open set $\mathbb{R}^+ \subset \mathbb{R}_{\ge 0}$ along each $u_i$. One could go in various directions with this: replace $\Sigma$ or $\mathbb{R}_{\ge 0}$ with another rig or algebra or something. Or perhaps allow $\mathbb{R}_{\ge 0}^n$ or some other sort of normed space with 'nice' properties. Has anyone seen analogues of this setup with coprobes determining what 'local' means? I mean in the sense that some notion of cover is extracted from coprobes.

   Something that just occurred to me is that usual open covers are given by things which are pullbacks of the open point in the Sierpinski space $\Sigma = \{0,1\}$ (this being a sort of strong subobject classifier - speaking heuristically), whereas partitions (not of unity, but of some arbitrary function $\sum_i u_i:X \to \mathbb{R}^+$) $\{u_i:X \to \mathbb{R}_{\ge 0}\}$ give rise to a cover by pulling back the open set $\mathbb{R}^+ \subset \mathbb{R}_{\ge 0}$ along each $u_i$.

   One could go in various directions with this: replace $\Sigma$ or $\mathbb{R}_{\ge 0}$ with another rig or algebra or something. Or perhaps allow $\mathbb{R}_{\ge 0}^n$ or some other sort of normed space with ’nice’ properties.

   Has anyone seen analogues of this setup with coprobes determining what ’local’ means? I mean in the sense that some notion of cover is extracted from coprobes.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 31st 2010
   • (edited May 31st 2010)
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   Author: Harry Gindi
   Format: MarkdownItex> Some 'work' for nice schemes, some 'work' for nasty schemes and so on. Except in this case instead of moving to covers that are more general (e.g. etale or Nisnevich) than the natural ones (Zariski or open immersions), we are moving to more specific covers, namely open covers induced by pulling back along certain coprobes (viz. partitions of unity). I don't really agree with your characterization of how these topologies are used. The Nisnevich topology is used pretty much exclusively for k-theory, the etale topology is used mainly for arithmetic geometry (connections between the etale fundamental group and Galois groups, better behavior over finite fields, etc). The fppf topology is really the natural topology for doing hardcore categorical stuff with algebraic spaces and stacks by a theorem of Artin. The fpqc topology is the universal effective epimorphic topology (only slightly coarser than the canonical (universally strict epimorphic covering families)). I'm not sure exactly what it's used for, but I'm reasonably certain it has nothing to do with changing covers to work with nasty spaces.

   Some ’work’ for nice schemes, some ’work’ for nasty schemes and so on. Except in this case instead of moving to covers that are more general (e.g. etale or Nisnevich) than the natural ones (Zariski or open immersions), we are moving to more specific covers, namely open covers induced by pulling back along certain coprobes (viz. partitions of unity).

   I don’t really agree with your characterization of how these topologies are used. The Nisnevich topology is used pretty much exclusively for k-theory, the etale topology is used mainly for arithmetic geometry (connections between the etale fundamental group and Galois groups, better behavior over finite fields, etc). The fppf topology is really the natural topology for doing hardcore categorical stuff with algebraic spaces and stacks by a theorem of Artin. The fpqc topology is the universal effective epimorphic topology (only slightly coarser than the canonical (universally strict epimorphic covering families)). I’m not sure exactly what it’s used for, but I’m reasonably certain it has nothing to do with changing covers to work with nasty spaces.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 31st 2010
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   Author: DavidRoberts
   Format: MarkdownItexWell, from the POV of spaces or manifolds, schemes are pretty nasty unless they are projective or affine varieties over $\mathbb{C}$. The whole point of introducing Grothedieck topologies was that the Zariski topology made a variety or scheme a bad space, and also there weren't enough Zariski covers for what needed to be done. I was under the impression that for Noetherian/complete/quasi-compact/separated/etc schemes, some of the more garden variety topologies were sufficient, and for really awful singular schemes, the more exotic topologies needed to be hauled out, but as you point out, it is more application-dependent, rather than property-dependent. I'm really trying to get a grip on this interplay between 'more' and 'less' covers for Top. Numerable covers are very nice (they admit a locally finite refinement, for a start), but for some spaces, they may not be enough (almost certainly won't be enough for non-normal spaces). Open covers - everyone knows about them - but for nasty nasty spaces, they may not be enough. For example, a covering space of a scheme in the etale topology looks very different to a covering space using the Zariski topology. Then for a topological space, I shouldn't expect to get something that looks like a covering space, except formally, when I pass to larger pretopologies. Attempts I've seen to consider generalisations of covering spaces so as to capture 1-dimensional homotopy information (by Fox, for example, using what he called 'overlays') only use local trivialisation using open covers, and just mess around with what the map does. I don't know if this is the only approach (it works to some extent, obviously) - why can't we use other pretopologies as algebraic geometry does? What does $\pi_1$ look like if we consider covering spaces which are locally trivial for the pretopology of open surjections? Or the profinite version, calculated by only using finite covering spaces?

   Well, from the POV of spaces or manifolds, schemes are pretty nasty unless they are projective or affine varieties over $\mathbb{C}$. The whole point of introducing Grothedieck topologies was that the Zariski topology made a variety or scheme a bad space, and also there weren’t enough Zariski covers for what needed to be done. I was under the impression that for Noetherian/complete/quasi-compact/separated/etc schemes, some of the more garden variety topologies were sufficient, and for really awful singular schemes, the more exotic topologies needed to be hauled out, but as you point out, it is more application-dependent, rather than property-dependent. I’m really trying to get a grip on this interplay between ’more’ and ’less’ covers for Top. Numerable covers are very nice (they admit a locally finite refinement, for a start), but for some spaces, they may not be enough (almost certainly won’t be enough for non-normal spaces). Open covers - everyone knows about them - but for nasty nasty spaces, they may not be enough. For example, a covering space of a scheme in the etale topology looks very different to a covering space using the Zariski topology. Then for a topological space, I shouldn’t expect to get something that looks like a covering space, except formally, when I pass to larger pretopologies. Attempts I’ve seen to consider generalisations of covering spaces so as to capture 1-dimensional homotopy information (by Fox, for example, using what he called ’overlays’) only use local trivialisation using open covers, and just mess around with what the map does. I don’t know if this is the only approach (it works to some extent, obviously) - why can’t we use other pretopologies as algebraic geometry does? What does $\pi_1$ look like if we consider covering spaces which are locally trivial for the pretopology of open surjections? Or the profinite version, calculated by only using finite covering spaces?

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 31st 2010
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   Author: Harry Gindi
   Format: MarkdownItex> Why can't we use other pretopologies as algebraic geometry does? What does look like if we consider covering spaces which are locally trivial for the pretopology of open surjections? Or the profinite version, calculated by only using finite covering spaces? Well, the fact that AG (and in some sense DG) behave so well is that they are modeled on nice spaces, in a sense. Even better, schemes are modeled on quasicompact sober spaces (or commutative rings depending on your construction of Sch), which are relatively well-behaved (not quite compact Hausdorff, but compact T1 and sober). Whatever kinds of pathology you have with schemes and manifolds will be infinitely worse in Top, because most topological spaces are pretty nasty.

   Why can’t we use other pretopologies as algebraic geometry does? What does look like if we consider covering spaces which are locally trivial for the pretopology of open surjections? Or the profinite version, calculated by only using finite covering spaces?

   Well, the fact that AG (and in some sense DG) behave so well is that they are modeled on nice spaces, in a sense. Even better, schemes are modeled on quasicompact sober spaces (or commutative rings depending on your construction of Sch), which are relatively well-behaved (not quite compact Hausdorff, but compact T1 and sober). Whatever kinds of pathology you have with schemes and manifolds will be infinitely worse in Top, because most topological spaces are pretty nasty.

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 29th 2016
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   Author: Dmitri Pavlov
   Format: TextIs there any reference about the numerable site that is available on arXiv, MathSciNet, or Zentralblatt (i.e., not just the nLab)? (Context: I would like to use the numerable site as an example for the paper https://dmitripavlov.org/concordance.pdf The idea here is that the numerable site has all the properties one needs for the main classification theorem, so without any additional effort one pulls out various theorems about classification of numerable bundles etc.)

   Is there any reference about the numerable site
   that is available on arXiv, MathSciNet, or Zentralblatt (i.e., not just the nLab)?
   
   (Context: I would like to use the numerable site as an example for the paper
   https://dmitripavlov.org/concordance.pdf
   The idea here is that the numerable site has all the properties one needs
   for the main classification theorem, so without any additional effort one pulls
   out various theorems about classification of numerable bundles etc.)
7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 29th 2016
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   Author: DavidRoberts
   Format: MarkdownItex@Dmitri have a look in my TAC article. I plan to write a short note one day about different pretopologies on Top, and the cases where they give equivalent sites.

   @Dmitri have a look in my TAC article. I plan to write a short note one day about different pretopologies on Top, and the cases where they give equivalent sites.

8. • CommentRowNumber8.
   • CommentAuthorZhen Lin
   • CommentTimeMar 1st 2016
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   Author: Zhen Lin
   Format: MarkdownItexAre there any interesting pretopologies on $Top$ which are not $\kappa$-ary superextensive for some $\kappa$ (in the strong sense of being generated by the singleton covering families and $\kappa$-ary coproduct cocones)?

   Are there any interesting pretopologies on $Top$ which are not $\kappa$-ary superextensive for some $\kappa$ (in the strong sense of being generated by the singleton covering families and $\kappa$-ary coproduct cocones)?

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 1st 2016
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   Author: DavidRoberts
   Format: MarkdownItex@Zhen Lin - not that I know of, as long as you mean to include the case when we replace $\kappa$-ary by 'small' I could probably construct pretopologies that give rise to equivalent sheaf toposes that aren't superextensive, but I don't know how intrinsically interesting they are.

   @Zhen Lin - not that I know of, as long as you mean to include the case when we replace $\kappa$-ary by ’small’

   I could probably construct pretopologies that give rise to equivalent sheaf toposes that aren’t superextensive, but I don’t know how intrinsically interesting they are.

10. • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeMar 1st 2016
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    Author: Zhen Lin
    Format: MarkdownItexI don't know of many pretopologies myself, let alone the uses of them, so I could hardly judge whether one is interesting or not. Here's a curious observation: because the class of proper surjections is closed under infinitary coproduct, there is an infinitary superextensive pretopology in which $\{ (U_i, x_i) : i \in I \}$ is in the pretopology if $\coprod_{i \in I} U_i \to X$ is a proper surjection. This feels wrong somehow, because proper surjections should go with finite covering families. What is true is that there is a certain finite character built into this pretopology: since the fibres of a proper map are compact, if $\coprod_{i \in I} U_i \to X$ is a proper surjection, then each point of $X$ is covered by only finitely many of the $(U_i, x_i)$; so I suppose one might call it locally finite...

    I don’t know of many pretopologies myself, let alone the uses of them, so I could hardly judge whether one is interesting or not.

    Here’s a curious observation: because the class of proper surjections is closed under infinitary coproduct, there is an infinitary superextensive pretopology in which $\{ (U_i, x_i) : i \in I \}$ is in the pretopology if $\coprod_{i \in I} U_i \to X$ is a proper surjection. This feels wrong somehow, because proper surjections should go with finite covering families. What is true is that there is a certain finite character built into this pretopology: since the fibres of a proper map are compact, if $\coprod_{i \in I} U_i \to X$ is a proper surjection, then each point of $X$ is covered by only finitely many of the $(U_i, x_i)$; so I suppose one might call it locally finite…

11. • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 1st 2016
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    Author: Dmitri Pavlov
    Format: Text@DavidRoberts: Thanks for the reference. I would be quite interested in reading your note when it comes out.

    @DavidRoberts: Thanks for the reference. I would be quite interested in reading your note when it comes out.