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Roman T

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 1st 2021

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 18th 2021

In the formulation of Quillen Theorem A, should it not say “initial (∞,1)-functor” instead of “cofinal (∞,1)-functor”? The page initial functor seems to use the same comma category…

• CommentRowNumber4.
• CommentAuthorHurkyl
• CommentTimeDec 19th 2021
• (edited Dec 19th 2021)

Initial and cofinal mean the same thing. Unless you’re using the other convention where cofinal means final.

That said, in my opinion using “initial” and “final” is the preferable convention, for no deeper reason that you don’t use the word “cofinal” and force the reader to figure out which of the two opposite conventions you’re using.

I went to check Higher Topos Theory (which uses the convention that “cofinal means final” – that is, cofinal functors relate to colimits) to make sure the theorem is stated correctly. This is theorem 4.1.3.1. $(d \downarrow F)$ is used in the version for final functors, so the $(F \downarrow d)$ at the nLab page should be the version for initial functors.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 19th 2021

Initial and cofinal mean the same thing.

What book or published article uses “cofinal” to mean “initial”?

• CommentRowNumber6.
• CommentAuthorHurkyl
• CommentTimeDec 19th 2021
• (edited Dec 19th 2021)

Cisinski’s “Higher Categories and Homotopical Algebra”, Definition 4.4.13 defines a morphism of simplicial sets $X \to Y$ to be cofinal iff the opposite morphism is final.

(and to make it clear it’s not using “final = initial” that I see mentioned in a nLab article, which I’ve not actually seen before, Cisinski defines a “final object” to be a final functor $\Delta^0 \to X$ and an “initial object” to be a cofinal functor $\Delta^0 \to X$)

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 19th 2021

Re #6: Thanks, I did not know about this. This is pretty awful! I think the “cofinal” terminology should be retired immediately.