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I can't edit the nlab at the moment, so I'll reply here. I think your construction AllButChosen(C) is just . If we assume choice in the meta-logic (this is needed to choose pullbacks of categories, which to my mind is not as strong as Choice, because there are constructions for ordered pairs of things, given certain foundations like ZF, even if they are not unique), then is a bicategory, but otherwise it is an anabicategory (=category enriched in ). In the former case, AllButChosen(C) is an ordinary category, and the functor AllButChosen(-) is . If we do away with choice altogether (and thus have the latter case of the aboce two options), then this really lands in $Cat_{ana}$$ again. This is a good idea, though.
You should add an entry for that terminology if possible. Or at least make a stub and post it on the n-forum so someone else can take care of it.
It may be a good idea to have a link to a page on cliques in graph theory. there is a Wikipedia page
http://en.wikipedia.org/wiki/Clique_%28graph_theory%29
which may do.
I edited "clique" a bit -- Makkai calls these "anaobjects" although I think "clique" is a good word.
I didn't notice whether this happened recently, but I don't like having the page anafunctor talk about functors being "k-surjective for all k." Right now the page is just about anafunctors between ordinary (internal) 1-categories, so it suffices to say "fully faithful and essentially surjective," which I think is more friendly to a lot of people. If we want to write about anafunctors between higher categories, great, but let's do it on a different page, or a special section at the end of that page.
I didn't notice whether this happened recently, but I don't like having the page anafunctor talk about functors being "k-surjective for all k."
I am too lazy to check, but if this is an old leftover from an edit I made, feel free to remove it. I agree that the page should concentrate on the 1-categorical case.
I fixed it.
Added the definitions of full and faithful anafunctors and composition of anafunctors, and some examples, including the promised inverses of fullly faithful essentially surjective anafunctors.
I fixed the syntax for referring back to an earlier section (on how to do this, see here).
Thanks! I somehow managed to get it the wrong way round without noticing…
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