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

- Jiří Adámek, Lurdes Sousa,
*A Finitary Adjoint Functor Theorem*, arXiv.

have copied the reference also to *Hans Porst*

Added a reference to Porst’s recent survey paper.

]]>none of the occurrences of “continuous functor” or “cocontinuous functor” here were hyperlinked. have changed that

]]>Add early reference for enriched adjoint functor theorem.

]]>Mention a generalisation of the AFT.

]]>Strengthened the first of the two statements for adjoint functors in the locally presentable case.

]]>Added a reference to

- Duško Pavlović,
*On completeness and cocompleteness in and around small categories*, APAL**74**(1995) pp.121-152.

[ 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.

]]>Changed notation for presheaves

Bartosz Milewski

]]>Further explanation of syntax

]]>Explained the non-standard notation for the limit.

Bartosz Milewski

]]>Added an adjoint functor theorem for cocomplete categories.

]]>Change notation in the statement of the theorem to match its proof (the functor is $R:C\to D$ instead of $G:D\to C$).

]]>Clarified the language in another relevant spot (where a counterexample was given).

]]>Clarified some language in the statements that characterize adjoints between locally presentable categories, in response to a comment made by user Hurkyl in another thread (here).

]]>Thanks.

]]>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.)

]]>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.

]]>