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-categories 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 complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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-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

    started a disambiguation page basis

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 25th 2010

    Disambiguated the disambiguation page basis :)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010
    • (edited May 25th 2010)

    Thanks.

    The disambiguation of “base” is something I am thinking actually the title of that entry might want to reflect.

    The disambiguation of “basis of a topology” is not so much one, is it? A basis of the Grothendieck topology of a category of open subsets of some topological space should correspond to a basis for the topology of that space.

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 25th 2010

    Replaced “basis of a vector space” with “basis of a free module”.

    • CommentRowNumber5.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 25th 2010

    I wrote up a page basis of a free module.

    • CommentRowNumber6.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 25th 2010

    Cleaned up the formatting a little.

    It’s a minor point, but if characterising a basis of a free module as an isomorphism between it and a “standard” free module, then I would write the isomorphism in the other direction; namely, as R[B]MR[B] \to M. I much prefer mapping out of free things than mapping in to them, even if they are isomorphisms!

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2010

    Yes, I think the titles of the pages base and basis for a topology are pretty confusing. I would maybe call them “base for a topology” and “basis for a Grothendieck topology” or something.

    Actually, is “basis for a (Grothendieck) topology” an unintentional duplication of Grothendieck pretopology that should be merged?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    Mike,

    I see, you are right. So I created quite some mess. Sorry. Will try to clean it up later.

    • CommentRowNumber9.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 25th 2010
    • (edited May 25th 2010)

    My wife just looked over my shoulder and wants to know why we’re not using the word “clarification”, seeing as that’s actually an English word (unlike “disambiguation”).

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2010

    “Disambiguation” is so an English word. It is also more specific than “clarification;” one can clarify things in many ways, but one disambiguates them particularly by removing a source of ambiguity.

    • CommentRowNumber11.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 26th 2010

    Andrew’s wife is probably referring to the famous essay by Orwell, “Politics and the English Language”.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2010

    I like that essay quite a lot, but why is Andrew’s wife “probably referring” to it?

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 26th 2010

    @Urs #3

    actually a basis for a topology is only a coverage of the category 𝒪\mathcal{O} of open sets of a space. A basis for a Grothendieck topology (a pretopology) on 𝒪\mathcal{O} is the usual thing: a cover of an object is just an open cover of the open set.

    • CommentRowNumber14.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 26th 2010

    @Todd: I dunno, but “disambiguation” is one of those Latin/Greek monstrosities that Orwell talks about in the essay. I didn’t give any thought to whether or not “clarification” is similar in this respect, and I don’t know its etymology offhand, but if it is not a loan-word, then my comment might make sense.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2010
    • (edited May 26th 2010)

    Oh, I see. Yes, I believe “disambiguation” is a Greek-Latin hybrid (whereas “clarification” is not), so okay. It’s by no means clear that Andrew’s wife was thinking of the essay (or has even read it), but at least I see what you were driving at.

    I feel bad that a presumably offhand comment by Andrew’s wife is being subjected to such scrutiny, but FWIW I note that the earliest citation of “disambiguation” in the OED seems to be from 1827 (unless my worsening eyesight deceives me).

    Edit: I’m wrong! “ambi-” is Latinate, and it was silly of me to think otherwise. So scratch that.

    • CommentRowNumber16.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 26th 2010
    • (edited May 26th 2010)

    By “Greek/Latin”, I meant “derived from either greek or latin”, mainly transmitted through French loan words that came into the language through the Normans. Anyway, I was wrong, because “clarification” also comes from latin.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2010

    @David #13:

    Right, stupid me, I wasn’t thinking.

    I added this statement to the Examples-sections here and here.

    • CommentRowNumber18.
    • CommentAuthorzskoda
    • CommentTimeMay 26th 2010
    • (edited May 26th 2010)

    A basis of the Grothendieck topology of a category of open subsets of some topological space should correspond to a basis for the topology of that space.

    Really ? Not a prebasis ? I mean pullbacks are intersections. In prebasis you can use intersections as well; in basis of topology you can not. There is also a basis of neighborhood of a point (or fundamental system of neighborhoods or basis of local topology).

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 27th 2010

    @Zoran - see my #13. I agree that there is also the concept of a basis of neighbourhoods - it is my favourite approach to defining a topology! But I don’t know how to reconcile the Grothendieck topology approach and neighbourhood bases. The latter after all uses points of the space, and this is not a good notion except for sober spaces, because for a non-sober space the open neighbourhoods aren’t enough to find the point again.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 27th 2010

    And see my #17.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMay 27th 2010

    So to straighten this out once and for all, I now created an entry basis for a topology after all, which discusses the topological notion and highlights its relation to the topos-theoretic notion.

    • CommentRowNumber22.
    • CommentAuthorzskoda
    • CommentTimeMay 29th 2010

    I do not understand 19 and 20. I objected to base vs. prebasis/subbase (both notions from general topology) and you point me to the diuscussion on the comparison with Grothendieck topology. This has nothing to do with my objection. Namely if you generate by pullbacks/intersections that means not only unions then it is subbase in classical case, not base.

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)