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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…
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.
Initial and cofinal mean the same thing.
What book or published article uses “cofinal” to mean “initial”?
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 $\Delta^0 \to X$)
Re #6: Thanks, I did not know about this. This is pretty awful! I think the “cofinal” terminology should be retired immediately.
Added details of published version:
- David Roberts, Homotopy types of topological stacks of categories, New York Journal of Mathematics, Volume 30 (2024), 940-955, journal version, arXiv:2204.02778
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