Which maps of simplicial sets geometrically realize to fibrations? If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a Serre fibration, and indeed then necessarily a Hurewicz fibration since a Serre fibration between CW complexes is a Hurewicz fibration.  Some references are given in this answer.  However, the point of that answer is that $f$ being a Kan fibration is not a necessary condition for $|f|$ to be a Hurewicz fibration.  In particular, since geometric realization preserves pushouts and cofibrations, and the pushout of a cospan of Hurewicz fibrations over a fixed base is again a Hurewicz fibration if one map in the cospan is a cofibration, it follows that a similar pushout of Kan fibrations involving a cofibration of simplicial sets geometrically realizes to a Hurewicz fibration.  So my questions are:


*

*Is there an exact characterization of those maps of simplicial sets whose geometric realizations are (Serre, hence necessarily Hurewicz) fibrations?

*If that's too hard, can we at least characterize some class of such maps that's larger than the Kan fibrations, and in particular includes pushouts of Kan fibrations along cofibrations as above?


Of course there's a continuum in what counts as a "characterization", e.g. for question 2 we could just say "the closure of Kan fibrations under pushouts along cofibrations".  We could also close up under sequential colimits of cofibrations, since I believe Hurewicz fibrations are also closed under those.  But what I'd really like is an intrinsic description of such a class that "looks like a notion of fibration", e.g. it is given by some kind of lifting property inside simplicial sets.
 A: Similar results are discussed in Section 2.7 of my book with Waldhausen and Jahren: ``Spaces of PL Manifolds and Categories of Simple Maps'':
https://folk.uio.no/rognes/papers/aoms186-nocrop.pdf 
We say that a map $f : X \to Y$ of finite simplicial sets is `simple' if the geometric realization $|f| : |X| \to |Y|$ has contractible point preimages.  Simple maps are simple-homotopy equivalences.
Proposition 2.7.6 shows that a map $\pi : Z \to \Delta[q]$ (with $Z$ finite) realizes to a Serre fibration $|\pi| : |Z| \to |\Delta[q]| = \Delta^q$ if and only if certain natural maps
$$
g : Sd(\pi)^{-1}(\beta) \to Sq(\pi)^{-1}(\mu)
$$
are simple, for each face $\mu$ of the maximal face $\beta$ of $\Delta[q]$, which
in turn holds if and only if a certain natural map
$$
t : Sd(\pi)^{-1}(\beta) \times \Delta[q] \to Z
$$
is simple.  Basically this generalizes the result that $g : A \to B$ is simple if
and only if the projection $\pi : Mg \to \Delta[1]$ from the mapping cylinder $Mg
= A \times \Delta[1] \cup_A B$ to $\Delta[1]$ realizes to a Serre fibration.
By Lemma 2.7.12, a map $\pi : E \to B$ of finite simplicial sets, where $B$ is nonsingular (each nondegenerate simplex is embedded), realizes to a Serre fibration if and only if it has this property when restricted to each (nondegenerate) simplex of $B$.
I do not know whether all simplexwise Serre fibrations are Serre fibrations when the base $B$ is singular (= not nonsingular).
