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 definitions 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab nonassociative 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.
   • CommentAuthorTobyBartels
   • CommentTimeJun 18th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI got tired of making unmatched links to [[topological locale]] (aka [[spatial locale]], or [[locale with enough points]]), so I wrote a stub.

   I got tired of making unmatched links to topological locale (aka spatial locale, or locale with enough points), so I wrote a stub.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeJun 18th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexAlso [[locale of opens]]. Very stubby. I think that I really only wrote these so that I could put in a bunch of redirects for each, and head off anybody else writing them with too few redirects. (^_^)

   Also locale of opens. Very stubby.

   I think that I really only wrote these so that I could put in a bunch of redirects for each, and head off anybody else writing them with too few redirects. (^_^)

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 18th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI would be more inclined to call the page "spatial locale" -- I've encountered that term much more often. Plus "topological" as an adjective usually means that we've put an extra topology on something, so a "topological locale" sounds like an internal locale in Top. I'm also not really fond of calling it the "locale of opens" -- I would talk about the *frame* of opens, but the locale is "of" the same things that the space was. E.g. the locale underlying the space of real numbers is the locale of real numbers, not the locale of open subsets of real numbers -- the latter would be some sort of "hyperlocale" whose points are open sets.

   I would be more inclined to call the page “spatial locale” – I’ve encountered that term much more often. Plus “topological” as an adjective usually means that we’ve put an extra topology on something, so a “topological locale” sounds like an internal locale in Top.

   I’m also not really fond of calling it the “locale of opens” – I would talk about the frame of opens, but the locale is “of” the same things that the space was. E.g. the locale underlying the space of real numbers is the locale of real numbers, not the locale of open subsets of real numbers – the latter would be some sort of “hyperlocale” whose points are open sets.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeJun 18th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex>Plus "topological" as an adjective usually means that we've put an extra topology on something, so a "topological locale" sounds like an internal locale in Top. Well, yeah, but the same is true of ‘spatial’. The term ‘topological’ here is acting in same way as ‘localic’ in ‘[[localic topos]]’. I don't like ‘spatial’ because a locale *is* a kind of space anyway! (But [[spatial locale]] redirects, of course.) The term ‘topological’, however, is well established to mean a topological space in the sense of Bourbaki, as when (for example) one says that a [[convergence space]] is ‘topological’. >I'm also not really fond of calling it the "locale of opens" -- I would talk about the frame of opens, but the locale is "of" the same things that the space was. Gosh, you're right! I was a little unhappy with that name on the grounds that *every* locale is a locale of opens; this is really just the locale of opens *in a topological space*. But it's worse than that; every locale is *really* a locale of *points*! (Just like a topological space is a space of points.) So what is the correct term here? For the locale I mean, not the frame. I suppose that one might say ‘topological locale’ (or ‘spatial locale’) again here, but what is one supposed to say when given a topological space $X$ and wanting to speak of the locale whose underlying frame is the frame of open subspaces of $X$? Somehow I have not picked up that terminology.

   Plus “topological” as an adjective usually means that we’ve put an extra topology on something, so a “topological locale” sounds like an internal locale in Top.

   Well, yeah, but the same is true of ‘spatial’. The term ‘topological’ here is acting in same way as ‘localic’ in ‘localic topos’. I don’t like ‘spatial’ because a locale is a kind of space anyway! (But spatial locale redirects, of course.) The term ‘topological’, however, is well established to mean a topological space in the sense of Bourbaki, as when (for example) one says that a convergence space is ‘topological’.

   I’m also not really fond of calling it the “locale of opens” – I would talk about the frame of opens, but the locale is “of” the same things that the space was.

   Gosh, you’re right! I was a little unhappy with that name on the grounds that every locale is a locale of opens; this is really just the locale of opens in a topological space. But it’s worse than that; every locale is really a locale of points! (Just like a topological space is a space of points.)

   So what is the correct term here? For the locale I mean, not the frame. I suppose that one might say ‘topological locale’ (or ‘spatial locale’) again here, but what is one supposed to say when given a topological space $X$ and wanting to speak of the locale whose underlying frame is the frame of open subspaces of $X$? Somehow I have not picked up that terminology.

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeJun 19th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexOK, I've reworked [[topological locale]] and [[locale of opens]] into a new version of [[topological locale]]. This does not use the terminology ‘locale of opens at all’, but instead introduces the notation $\Omega(-)$ for the concept. I have also written [[frame of opens]] (which I made out of the old [[locale of opens]], although that name now redirects to [[topological locale]], since that is the page that discusses the concept that I had earlier had at the badly named page [[locale of opens]]). And I am writing [[space of points]] as well.

   OK, I’ve reworked topological locale and locale of opens into a new version of topological locale. This does not use the terminology ‘locale of opens at all’, but instead introduces the notation $\Omega(-)$ for the concept. I have also written frame of opens (which I made out of the old locale of opens, although that name now redirects to topological locale, since that is the page that discusses the concept that I had earlier had at the badly named page locale of opens). And I am writing space of points as well.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJun 19th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> the same is true of ‘spatial’. Not to me. I've never heard people talk about, say, a "spatial group" or a "spatial category" with the same meanings as the established terms "topological group" and "topological category." But I can live with "topological locale." I don't know of a standard term for "the locale underlying a topological space." I suppose if we identify spatial locales with sober spaces, as one might argue is not unreasonable, then it would be the same as the soberification, but that'd be misleading here.

   the same is true of ‘spatial’.

   Not to me. I’ve never heard people talk about, say, a “spatial group” or a “spatial category” with the same meanings as the established terms “topological group” and “topological category.” But I can live with “topological locale.”

   I don’t know of a standard term for “the locale underlying a topological space.” I suppose if we identify spatial locales with sober spaces, as one might argue is not unreasonable, then it would be the same as the soberification, but that’d be misleading here.