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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMay 3rd 2021
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWhen we say > A [[category]] $C$ is **cofinally small** if there is a [[small category]] $C_0$ and a [[cofinal functor]] $C_0 \to C$. does "cofinal functor" mean "final functor" or "co-(final functor)" i.e. "initial functor"? The only links to this page I can find are from [[calculus of fractions]], where my guess would be that what's meant is "initial functor", and from [[ind-object]], where my guess would be that what's meant is "final functor". So I don't know. <a href="https://ncatlab.org/nlab/revision/diff/cofinally+small+category/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/cofinally+small+category/2">v2</a>, <a href="https://ncatlab.org/nlab/show/cofinally+small+category">current</a>

   When we say

   A category $C$ is cofinally small if there is a small category $C_0$ and a cofinal functor $C_0 \to C$.

   does “cofinal functor” mean “final functor” or “co-(final functor)” i.e. “initial functor”?

   The only links to this page I can find are from calculus of fractions, where my guess would be that what’s meant is “initial functor”, and from ind-object, where my guess would be that what’s meant is “final functor”. So I don’t know.

   diff, v2, current

2. • CommentRowNumber2.
   • CommentAuthorRichard Williamson
   • CommentTimeMay 3rd 2021
   • (edited May 3rd 2021)
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   Author: Richard Williamson
   Format: MarkdownItexI think that 'final functor' is the almost universal meaning in the literature; in fact, whilst I'm writing this quickly, I would have thought this is also what is meant at [[calculus of fractions]]? What a terribly confusing piece of terminology!

   I think that ’final functor’ is the almost universal meaning in the literature; in fact, whilst I’m writing this quickly, I would have thought this is also what is meant at calculus of fractions?

   What a terribly confusing piece of terminology!

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 3rd 2021
   • (edited May 3rd 2021)
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   Author: Dmitri Pavlov
   Format: MarkdownItexThis was discussed here: <https://nforum.ncatlab.org/discussion/9054/final-functor/>. Apparently, Borceux's [[Handbook of Categorical Algebra]] and Lurie's [[Higher Topos Theory]] use “cofinal functor” for the concept that all other books call “final functor”.

   This was discussed here: https://nforum.ncatlab.org/discussion/9054/final-functor/.

   Apparently, Borceux’s Handbook of Categorical Algebra and Lurie’s Higher Topos Theory use “cofinal functor” for the concept that all other books call “final functor”.