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nLab > Latest Changes: filtered category

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  1.  
    • 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-filteredness for the finite regular cardinal 2, 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.

    I had though that κ was a regular cardinal, whilst ω was an ordinal. Was ℵ0 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-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 = ℵ0-compact’. The question seems to hinge, as well, on whether certain < are ≤ or not!

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2014

    Right, cardinals are by definition (in most approaches used today) certain types of ordinals, and ω happens to be that type of ordinal. It would have been okay to say ℵ0-compact objects, but it’s much more usual to see reference in the literature to ω-compact objects, which are the same as compact objects.

    The condition for a category C to be κ-filtered is that for every diagram D→C where D is of size <κ, there is an extension D+→C where D+ is obtained by adjoining a terminal object to D. So ω-filtered means that every diagram in C of size <ω, i.e., every finite diagram D→C, has such an extension; these are called just filtered categories. And so ω-compact has to do with 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 ∞-filtered categories.

    diff, v32, current

1 to 11 of 11

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