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nLab > Latest Changes: final functor

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  1. Added the property that final functors and discrete fibrations form an orthogonal factorisation system.

    diff, v22, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 28th 2019

    Borceux (Handbook of Categorical Algebra, Volume 1, Definition 2.11.1) defines final functors as those functors for which restriction preserves limits, which is the exact opposite of what the nLab says (i.e., restriction along final functors preserves colimits).

    Is this just Borceux’s idiosyncrasy? His books generally appear to use conventional terminology. Shouldn’t this be mentioned in the article?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 28th 2019

    Added alternative meaning of “final” to the Idea section. (Making explicit something left somewhat implicit before in the mention of the alternative meaning of “cofinal”.)

    diff, v23, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 3rd 2019

    The new version is confusing: “final” appears as a name for two opposite notions!

    Maybe we should offer some guide to the terminology. I already mentioned Borceux’s book terminology: cofinal for colimits, final for limits. Mac Lane’s terminology: final for colimits, initial for limits. What other canonical sources should be considered?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2019

    I think Borceaux is the odd one out. We can mention his terminology, but we shouldn’t suggest that its use be continued.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2019

    More extensive and less ambiguous advice about terminology.

    diff, v24, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 3rd 2019

    Added a quote from Johnstone’s Elephant

    diff, v25, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 3rd 2019

    Added another reference point: Lurie’s Higher Topos Theory uses “cofinal”.

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2021

    added (here) statement of the example of the final functor

    ([1]⇉d 0d 1[0])→Δ op \big( [1] \underoverset {d_0} {d_1} {\rightrightarrows} [0] \big) \;\xrightarrow{\;\;\;\;}\; \Delta^{op}

    diff, v30, current

    • CommentRowNumber10.
    • CommentAuthorJoshua Meyers
    • CommentTimeDec 29th 2021

    Made a link to a new page “comprehensive factorization system”.

    diff, v32, current

  11. I think in the definition of final functor, there is a typo: “F:C→DF : C \to D is final if for every object d∈Dd \in D the comma category (d/F)(d/F) is……” should be: F:C→DF : C \to D is final if for every object c∈Cc \in C the comma category (c/F)(c/F) is……

    Jishen Du

    diff, v34, current

  12. Sorry, should not edit in the beginning. My bad

    Jishen Du

    diff, v34, current

    • CommentRowNumber13.
    • CommentAuthorHurkyl
    • CommentTimeJun 28th 2022
    • (edited Jun 28th 2022)

    (d/F)(d/F) is correct; (c/F)(c/F) doesn’t type check. In particular, (d/F)(d/F) is the comma category from the functor 1→D1 \to D that picks out dd to the functor F:C→DF : C \to D.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2023

    added a brief mentioning (here) of the discussion of finality with respect to weighted limits by:

    This would be useful to have a citable reference for.

    diff, v40, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2023

    This would be useful to have a citable reference for.

    Oh, I see now that this can essentially be cited from Kelly §4.5. Will make an edit..

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2023

    added mentioning of the characterization of final functors between groupoids (here)

    and added pointer to:

    diff, v42, current

    • CommentRowNumber17.
    • CommentAuthorBryceClarke
    • CommentTimeJan 18th 2024

    Added some more properties of final functors (with arise from being the right class of an orthogonal factorisation system), as well as the example of final functors from a discrete category.

    diff, v44, current

    • CommentRowNumber18.
    • CommentAuthorGnampfissimo
    • CommentTimeJul 14th 2024

    In the characterization (proposition 3.1), item 4, shouldn’t the target of GG be Set opSet^op instead of SetSet? Since there is no proof or reference given I can’t be sure, but I see no reason to believe that we can reconstruct the connectedness condition by limits instead of colimits in SetSet.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2024

    this is taken from Kashiwara & Schapira 2006, Prop. 2.5.2

    (which was still more visibly the reference given when this was added in revision 8 in Jan 2010)

    I have made it clearer now

    diff, v45, current

    • CommentRowNumber20.
    • CommentAuthorvarkor
    • CommentTimeJul 19th 2024

    Expanded the reference to Kelly.

    diff, v46, current

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