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• CommentRowNumber1.
• CommentAuthorHarry Gindi
• CommentTimeMay 12th 2018

A while back, a friend of mine mentioned that Lax n-functors between strict n-categories are related to maps between the associated (unmarked) Street Nerves in the 2-categorical case at least. Is the correspondence bijective? If it is, does it generalize to the case n>2? If it isn’t, can we identify a particular subset of simplicial maps between the nerves that arise from lax functors in the case n=2? In the case n>2?

If so, I think it would be nice to add this to the nLab. The only source I have found on this is a paper of Jonathan Chiche, but he only seems to use the fact that lax 2-functors give rise to maps on the Street nerves in order to define weak equivalences (a lax functor is called a weak equivalence if the induced map on Street nerves is a weak equivalence of simplicial sets).

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMay 12th 2018

I think the correspondence is bijective on normal lax 2-functors. I believe Nick Gurski proved this, but I don’t know whether he ever published it. For $n\gt 2$ I don’t even know exactly what a “lax $n$-functor” is. Is it lax at all dimensions simultaneously? How do the constraints interact?

• CommentRowNumber3.
• CommentAuthorHarry Gindi
• CommentTimeMay 12th 2018
• (edited May 12th 2018)

I guess I meant normal lax functors, since this is, I believe, what Chiche was using.

I’ll send Nick an e-mail maybe.Thanks!