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 bundle bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched etcs fibration 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 k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure measure-theory modal modal-logic model model-category-theory 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 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.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 23rd 2019

    Created the page.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 23rd 2019

    Changed the wording slightly so that “element of a locale” refers to element of the corresponding frame.

    I seem to have seen this language use recently. Personally I think it might be less confusing to say “opens of a locale” instead of “elements”.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2019

    In older books and papers (and, perhaps, some recent work sticking to older terminology) one often finds the word “locale” used to mean what we nowadays call “frame”, so that an element of the locale is the same as an element of the frame. But I would argue that the modern perspective, whereby a locale means an object of Frm opFrm^{op} instead, means that “element of a locale” doesn’t really mean anything, since Frm opFrm^{op} doesn’t have a canonical forgetful functor to SetSet. “Opens of a locale” seems a better term to me.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2019

    Change to use “open of a locale” and write O(L)O(L) for the frame of opens of a locale LL.

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 25th 2019
    • (edited Jul 25th 2019)

    since Frm^op doesn’t have a canonical forgetful functor to Set

    Actually, Frm^op does have a canonical forgetful functor to Set: just take the right adjoint map of posets, which is guaranteed to exist.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 25th 2019

    I suppose you could call that canonical, but it doesn’t have most of the usual properties of forgetful functors. Or, put differently, defining “a locale is a complete lattice satisfying the infinite distributive law of finite meets over arbitrary joins” and “a locale morphism is a monotone map preserving arbitrary meets whose left adjoint also preserves finite meets” is a pretty ad hoc definition of the category LocLoc, and doesn’t to my mind justify calling the elements of a frame “elements” of its corresponding locale.

  1. By analogy we should also refer to a “sheaf of a Grothendieck topos” rather than an “object of a Grothendieck topos”. But I suspect that too many people identify toposes with their category of sheaves for that change of nomenclature to stick.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJul 27th 2019

    Well, we don’t say “element of a Grothendieck topos”. A topos has both “points” and “objects”, generalizing how a locale has “points” and “opens”.

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)