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I fixed a trivial typo in adjoint functor theorem but left wondering about this:
… the limit
$L c := \lim_{c\to R d} d$over the comma category $c/R$ (whose objects are pairs $(d,f:c\to R d)$ and whose morphisms are arrows $d\to d'$ in $D$ making the obvious triangle commute in $C$) of the projection functor
$L c = \lim_{\leftarrow} (c/R \to D ) \,.$
I don’t really understand this (and while I could figure it out, it’s probably not good to make readers do so). At first it sounds like someone is saying “the limit $L c$ over the comma category of the projection functor $L c$”, which would be circular. But it must be that both formulas are intended as synonymous definitions of $L c$. At that point one is left wondering why one has a backwards arrow under it and the other does not. I guess old-fashioned people prefer writing limits with backwards arrows under them, so someone is trying to cater to all tastes? I think it’s better in this website to use $lim$ and $colim$ for limit and colimit.
I could probably guess how to fix this, but I won’t since I might screw something up.
I went ahead and made some changes per your comment. See if that looks better. (I think I’d try a different explanation if I were writing this – or writing this today in case I was the one who wrote that then! – but never mind.)
Thanks.
[ since Bartosz emailed me about this: ]
The above edits concern the section Examples – In presheaf categories.
Bartosz wanted to make notationally explicit the Yoneda embedding in the various formulas shown there. I have now touched the section myself, added the remaining instances of the Yoneda embedding; and also made some further cosmetic changes to the typesetting, such as height-aligned parenthesis etc.
Added a reference to
have copied the reference also to Hans Porst
Added:
A stronger version for finitary functors between locally finitely presentable categories whose domain is ranked, requiring only the preservation of countable limits for the existence of a left adjoint, is discussed in
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