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nLab > Latest Changes: locally finite cell complex

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021

    a stub, just for completeness

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021

    renamed to more generic title

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021
    • CommentRowNumber4.
    • CommentAuthorDean
    • CommentTimeFeb 21st 2025
    A CW complex is locally finite if each cell is disjoint from all but finitely many cells of X. (from Ross Geoghegan's book). No doubt a cell has infinitely many points.

    So, somehow each of which must somehow cease to be involved in attaching maps fₙ : Sⁿ → Xₙ of high enough n?

    (I'll be in touch shortly with progress on my braid group project)
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2025

    Hi Dean,

    are you wondering why “each cell is disjoint from all but finitely many cell” is equivalent to “each point is contained in a finite number of cells”?

    This is unrelated the the infiniteness of points in a cell, as it refers to each single one of them being contained in only a finite number of cells.

    • CommentRowNumber6.
    • CommentAuthorDean
    • CommentTimeFeb 21st 2025
    Sorry, the infinite cardinality of each cell was indeed unrelated.

    I was trying to construct a CW-replacement functor which is simultaneously an endofunctor of locally compact locales, but it seems there is only one which matches on πₖ for 0 ≤ k ≤ n (weak equivalence of n-truncations).

    Here are some facts I collected to that end:

    - Filtered colimits do not preserve locally compactness:

    https://ncatlab.org/nlab/show/locally+compact+topological+space#categorytheoretic_properties

    - From that and the facts in the page I have convinced myself that the CW-replacement functor in which one considers an attaching map for each element of Cellₙ := [Dⁿ,X] is demonstrably not compact because of how the continuous functions Dⁿ⁺¹ → Dⁿ entail that the resulting CW-complex is not locally finite.

    - The result about locally finite CW-complexes (that they are precisely the locally compact ones) each Replₙ (X) = Xₙ (functorial n-skeleton) is not difficult to make into an endofunctor of locally compact spaces.

    - All locally compact locales are spatial

    One thing I still do not understand is the possibility of having πₖ(Replₙ X) = 0 for k > n, which also seems impossible.

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