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• CommentRowNumber1.
• CommentAuthorAdam Chalcraft
• CommentTimeFeb 26th 2021
(This is my first foray into nLab, so sorry if I'm making elementary errors.)
In the definition of left adjoint of a functor U:C→ D, the claim is that it's a functor F:D → C s.t. ∃ natural transformations
ι:id_C → F;U
ϵ:U;F → id_D
But F;U is a morphism in D and U;F is a morphism in C.
Is something wrong here, have I misunderstood the notation F;U, is there a more general version of a natural transformation being used here, or what?
Thank you.
• CommentRowNumber2.
• CommentAuthorHurkyl
• CommentTimeFeb 26th 2021
• (edited Feb 26th 2021)

Some of the 2-category theory articles at nlab, I think, use the semicolon for horizontal composition: i.e. the functor $\hom(Y,Z) \times \hom(X,Y) \to \hom(X,Z)$.

So, $F;U$ just means the ordinary composition of the two functors.

Except… now that I’m looking at the article it looks like it’s the reversed convention? I.e. $\hom(X,Y) \times \hom(Y,Z) \to \hom(X,Z)$, so $F;U$ means the composite $UF$?

1. Gave the page more structure, corrected a few typos and notational confusions, and generally tried to improve it.

2. Also deleted a vague sentence in the introduction which struck me as more likely to be confusing/misleading than helpful.

• CommentRowNumber5.
• CommentAuthorRichard Williamson
• CommentTimeFeb 26th 2021
• (edited Feb 26th 2021)

Hi Adam, thanks very much for raising this! The page was rather a mess before, with all sorts of notational confusion and typos; I have now, as described in #3 and #4, edited the page. Hopefully things are now clear, but if not just let us know (and feel free to edit this or any other page if you see something you wish to correct or add!).

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