Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite 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 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

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.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 29th 2013
    • (edited Jun 29th 2013)

    In the article compactum, under Stone-Cech compactification, it is stated that the ultrafilter monad is commutative. I really don’t think that’s true (and it has nothing to do with βS\beta S being a compactum).

    To say β\beta is commutative is more or less saying that all the definable operations X JXX^J \to X (parametrized by elements θ\theta of the set βJ\beta J) on a β\beta-algebra XX are continuous. Explicitly, the associated operation is given by the composite

    X J=hom(J,X)hom(βJ,βX)hom(θ,1)hom(1,βX)βXαXX^J = \hom(J, X) \to \hom(\beta J, \beta X) \stackrel{\hom(\theta, 1)}{\to} \hom(1, \beta X) \cong \beta X \stackrel{\alpha}{\to} X

    where α:βXX\alpha: \beta X \to X is the algebra structure. I am reasonably confident all these operations are distinct even for a simple case like X=2X = \mathbf{2}, the two-point discrete space (edit: indeed, it may be shown that the operation 2 J2\mathbf{2}^J \to \mathbf{2} coincides with the ultrafilter θ\theta viewed as Boolean algebra map 2 J2\mathbf{2}^J \to \mathbf{2} – details available for anyone interested). For example, if J=J = \mathbb{N}, then there would be β\beta \mathbb{N} many such operations, which has cardinality 2 c2^c.

    On the other hand, there are only countably many continuous maps 2 2\mathbf{2}^{\mathbb{N}} \to \mathbf{2}. This is because 2 \mathbf{2}^{\mathbb{N}} is the Stone space attached to the free Boolean algebra Bool()Bool(\mathbb{N}) on countably many elements, and we retrieve a Boolean algebra BB from its Stone space Stone(B)Stone(B) by taking the set of continuous maps Stone(B)2Stone(B) \to \mathbf{2}, equipped with the pointwise Boolean algebra structure. In other words, the hom-set Top(2 ,2)Top(\mathbf{2}^{\mathbb{N}}, \mathbf{2}) is in natural bijection with the countable Boolean algebra Bool()Bool(\mathbb{N}).

    If there are no objections, I’ll go in and fix this.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 29th 2013

    I have now made some edits to the section on Stone-Cech compactification, and I made a new section on the category of compact Hausdorff spaces and its wonderful properties.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 30th 2013
    • (edited Jun 30th 2013)

    Giraud’s theorem also requires infinite coproducts to be disjoint and stable. Are they?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 30th 2013

    Oh, right (thanks). Oops. Well, I doubt it.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeJun 30th 2013

    If we follow the conventions in the Elephant, a pretopos only needs to be finitary-extensive.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 30th 2013

    @Zhen: yes, but I think Mike might have been alluding to my rash comment on Giraud’s theorem implying that the only obstacle to CompComp (or CHCH) being a Grothendieck topos is that it doesn’t have a small generating family.

    And I think I can turn my doubt in #4 to a positive “no”. If infinite coproducts were stable under pullbacks, then (for example) we’d have that X×:CHCHX \times -: CH \to CH preserves colimits (right?). But since CHCH is both total and cototal, this would be enough to conclude X×X \times - has a right adjoint (I think it’s totality that’s relevanthere, but never mind). But CHCH is not cartesian closed.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJun 30th 2013

    Right. A pretopos is finitely extensive, but the hypothesis of Giraud’s theorem is that the category is an infinitary-pretopos, which must by definition be infinitary-extensive.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2017

    I have edited the Idea-section at compactum slightly, for readability. Since “compact Hausdorff space” redirects to this entry, I think the Idea-section should be clear about this right away.

    The Idea section also has, and used to have, the following lines in it:

    One may consider the analogous condition for convergence spaces, or for locales. Even though these are all different contexts, the resulting notion of compactum is (at least assuming the axiom of choice) always the same.

    What is actually meant by this (“always the same”)? The entry never comes back to this statement. I suppose if a topological space is compact Hausdorff as a topological space, then it is also compact Hausdorff when regarded as a locale (?). But the claim is not that all compact Hausdorff locales are spatial locales coming from compact Hausdorff spaces, or is it?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2017

    But the claim is not that all compact Hausdorff locales are spatial locales coming from compact Hausdorff spaces, or is it?

    Oh, I see from locally compact locale that it is.