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Recall the following analogy

Serre fibrations : Kan fibrations

in spaces and simplicial sets respectively, related by the singular simplices functor and geometric realisation. There are other sorts of fibrations on each side. Can anyone fill in the following analogies ...

quasifibrations : ??

?? : inner fibrations

... if they do indeed exist. I'm not pedantic about using the specific adjunction $S\dashv |-|$.

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The analogies I'm after may not be symmetric... –  David Roberts Jul 16 '12 at 6:19
Peter's and Charles's answers are excellent. It seems to me that to have a topological analogue of inner fibrations you'd need to use some sort of directed spaces. I thought about something like that for a while once, but didn't really get anywhere. –  Mike Shulman Jul 16 '12 at 19:48

2 Answers 2

I have a preprint http://arxiv.org/abs/math/9811038 where I develop a bit about the simplicial analogue of quasifibrations; they are called "sharp maps" there. (There is actually a kind of local-to-global principle for these too.)

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Quasifibrations can be defined simplicially since one can develop the analogue of the mapping class fibration and ask for an equivalence from the fiber to the homotopy fiber. (Edit: on second thought this isn't quite that obvious.) It is pointless to do so since the value of quasifibrations is their local to global topological characterization, which makes it easy to recognize them when you see them. Of course inner fibrations make no sense topologically, since all topological horns are created equal. It is the valuable idiosyncrasies of different Quillen equivalent categories that make it worthwhile having them.

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