One of the most powerful results in ordinary category theory is the Yoneda lemma, and so the following question seems natural:
Is there an analogue of the Yoneda lemma for (weak) $n$-categories?
I am aware of a bicategory version (a nice proof of which can be found here), as well as a tricategory version (c.f. $\S$8 here), and searching around leads almost immediately to the nLab page for $(\infty,1)$-category version, but how about other cases?
Of course, the general definition of a weak $n$-category isn't exactly settled, so I realize that a proof in the general case isn't likely to exist at the moment, but is anyone aware of work done in other lower-dimensional cases?
Just in case anyone happens across this, just today there was a new paper posted to arXiv today proving the Yoneda lemma for $(\infty,1)$-categories using the language of complete Segal spaces. This is interesting, since there is a more natural generalization of the language of complete Segal spaces to $(\infty,n)$-categories, so perhaps this will provide an important step in Yoneda for such categories….