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

• 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”.)

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 2nd 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

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 3rd 2019

Added a quote from Johnstone’s Elephant

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 3rd 2019

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

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTime5 days ago

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

$\big( [1] \underoverset {d_0} {d_1} {\rightrightarrows} [0] \big) \;\xrightarrow{\;\;\;\;}\; \Delta^{op}$
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