Not signed in (Sign In)

Start a new discussion

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics comma complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 infinity integration integration-theory k-theory kan lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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 subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    created basis for a topology and linked to it with comments from coverage and, of course, Grothendieck topology

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    added the link to basis for a topology in the corresponding stub-subsection of the entry base.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeMay 25th 2010

    There is also a calssical notion of a base of topology where topology is in the sense of classical topological structure.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMay 25th 2010

    There is also a classical notion of a base of topology where topology is in the sense of classical topological structure.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010
    • (edited May 25th 2010)

    Okay, I merged what used to be “basis for a topology” into Grothendieck pretopology, putting suitable redirects and all (even successfully clearing the cache, now how is that?).

    I added a mention of the counterexample of the good open cover coverage, which is not a pretopology.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2010

    Thanks. Should it be obvious to me that good open covers are even a coverage?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    Should it be obvious to me that good open covers are even a coverage?

    Every open cover may be refined by a good open cover, by choosing a good open cover of each of the open subsets that are part of the open cover.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2010

    Are you assuming some sort of local contractibility of the spaces involved?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    Are you assuming some sort of local contractibility of the spaces involved?

    Ah, yes. I mean, I am thinking of this as a coverage on CartSp, even.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    Well, let’s see: at coverage in the Examples-section currently good open covers is listed as a coverage on Diff. I think that’s correct, but let me know if I am overlooking something.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2010

    Okay, I believe it on Diff; I was confused because good open cover referred to arbitrary topological spaces.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    Right. I hope it doesn’t claim that on Top good open covers form a coverage, just that there is a notion of good open cover for any topological space.

    I am in the process of checking some details. I’ll try to write a more comprehensive account on where good open covers form a coverage and where not a little later.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 26th 2010
    • (edited May 26th 2010)

    Can we get away with knowing that the inclusion map of each open is null-homotopic, instead of the open being contractible? This property is enough for applications like classifying locally homotopy trivial fibrations (cf Wirth-Stasheff’s JHRS paper)

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2010
    • (edited May 26th 2010)

    null-homotopic, instead of the open being contractible?

    So you define contractible here not as null-homotopic?

    What can we say about this here: for XX a locally contractible topological space, let OpC(X)Op(X)OpC(X) \subset Op(X) be the full subcategory of open subsets on those that are contractible. Both are equipped with the coverage of good open covers.

    Now, let’s see, Sh(OpC(X))Sh(OpC(X)) is still equivalent to Sh(Op(X))Sh(Op(X)), I suppose, and…

    • CommentRowNumber15.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 26th 2010

    There’s a difference between a space being contractible and a map being null-homotopic. The inclusion of a contractible subset is null-homotopic, but there are null-homotopic inclusions without the subspace being contractible, for example if the ambient space is contractible - so the inclusion of a circle in the plane is null-homotopic.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2010

    Ah, I see, you are thinking of null homotopy in the ambient space.

    Hm, do we really not have terms to distinguish between these two notions at locally contractible space?

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeMay 26th 2010

    By analogy with “semi-locally simply connected” it seems that a natural phrase for admitting local opens whose inclusions are nullhomotopic would be “semi-locally contractible.”

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2010

    That sounds good. I have implemented that now.

    David, is that allright with you?

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 27th 2010

    Yeah it’s good. I had something there, but it was a bit sketchy. I tidied up a bit where my old terminology conflicted with semi-locally contractible.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 27th 2010

    Just put a bit more at locally contractible space, including an example from MO I just asked for.

    • CommentRowNumber21.
    • CommentAuthorzskoda
    • CommentTimeMay 27th 2010

    Well, Andrew, I dislike that you posted a link at ndigest to this page, as what I put in zoran: new entries are side topics which are not worthy to open a separate page, so it gives exactly opposite impression from what I am really focused on in nlab.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)