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nLab > Latest Changes: filtered category

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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2012

    I added a section to filtered category about generalized filteredness relative to a class of small categories, as studied by Adamek-Borceux-Lack-Rosicky, and mentioned that it yields a better notion of κ\kappa-filteredness for the finite regular cardinal 22, as pointed out by Zhen in another thread.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeJan 31st 2014
    • (edited Jan 31st 2014)

    Can I ask clarification on the sentence:

    The usual filtered categories are then the case κ=ω\kappa = \omega.

    I had though that κ\kappa was a regular cardinal, whilst ω\omega was an ordinal. Was ℵ 0\aleph_0 intended or have I missed something, (in which case I would suggest that an additional word or two would be useful). My reason for asking was that I have been looking for the precise relationship between compact object and κ\kappa-compact object, and the entries in the Lab do not give the relationship in simple terms (i.e. simple enough for me :-(). I presume ‘compact = ℵ 0\aleph_0-compact’. The question seems to hinge, as well, on whether certain < are ≤\leq or not!

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2014

    Right, cardinals are by definition (in most approaches used today) certain types of ordinals, and ω\omega happens to be that type of ordinal. It would have been okay to say ℵ 0\aleph_0-compact objects, but it’s much more usual to see reference in the literature to ω\omega-compact objects, which are the same as compact objects.

    The condition for a category CC to be κ\kappa-filtered is that for every diagram D→CD \to C where DD is of size <κ\lt \kappa, there is an extension D +→CD^+ \to C where D +D^+ is obtained by adjoining a terminal object to DD. So ω\omega-filtered means that every diagram in CC of size <ω\lt \omega, i.e., every finite diagram D→CD \to C, has such an extension; these are called just filtered categories. And so ω\omega-compact has to do with hom(c,−)hom(c, -) preserving filtered colimits.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeJan 31st 2014

    Thanks Todd. You confirmed what I had thought was in fact correct.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2014

    Since the entry apparently didn’t make this clear enough: Todd, might you have a minute to add some more explanation to the entry? So that the next reader will know for sure? That would be great.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2014

    I tried to say something extra at filtered category.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2014

    Thanks! Excellent.

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 1st 2014

    I second that!

  9. Removing redirects for cofiltered categories, etc, as I will create a separate page for these to spell out some details explicitly.

    diff, v28, current

    • CommentRowNumber10.
    • CommentAuthorBryceClarke
    • CommentTimeNov 8th 2023

    Replaced a broken reference link.

    diff, v31, current

    • CommentRowNumber11.
    • CommentAuthorvarkor
    • CommentTimeDec 9th 2023

    Mentioned ∞\infty-filtered categories.

    diff, v32, current

1 to 11 of 11

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