Analogues of fibrations

Recall the following analogy

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 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 at 19:48