# 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:

1. Is there an exact characterization of those maps of simplicial sets whose geometric realizations are (Serre, hence necessarily Hurewicz) fibrations?
2. 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.

• Are $X$ and $Y$ are Kan complexes (may simplify things)? quasicategories? general simplicial sets? If you don't want to assume that $X$ and $Y$ are Kan complexes, then there are various sufficient but not necessary conditions that people have considered -- cf Rezk's note or this question. This comes up e.g. in all kinds of delooping machine technology. Also do quasifbrations suffice? BTW why are Hurewicz fibrations closed under pushout along (Hurewicz?) cofibrations? – Tim Campion Jan 11 at 17:22
• @TimCampion Well, if there's a useful condition that includes assumptions on $X$ and $Y$, that would be something, although I'd prefer one that's only about the map. I don't see the relevance of either of your two links, can you explain? The closure of Hurewicz cofibrations under pushout along Hurewicz cofibrations is in the paper of Clapp linked to in the answer I linked to. And I don't understand your second comment; I didn't claim that every pushout of a Serre fibration along a cofibration is a Hurewicz fibration. – Mike Shulman Jan 11 at 18:14
• @TimCampion Ah, yes. Now I see how my statement was easy to misinterpret. I'll edit the question. – Mike Shulman Jan 12 at 0:39
• Probably I've been spouting nonsense. Anyway, now I'm trying to think of an explicit example of a map which is not a Kan fibration but whose realization is a Serre fibration. Is the quotient map out of $\Delta[2]$ which collapses the edge from $0$ to $1$ an example? – Tim Campion Jan 12 at 3:06
• If you want a simple explicit example of a map that is not a Kan fibration but whose realization is a Serre fibration, just take a simplicial set $X$ that is not a Kan complex, and map it to a point. – Dan Ramras Jan 12 at 21:28

Similar results are discussed in Section 2.7 of my book with Waldhausen and Jahren: Spaces of PL Manifolds and Categories of Simple Maps'':

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).

• Thanks! That's interesting, but the restriction to finite simplicial sets and nonsingular bases is pretty drastic compared to what I'm hoping for. – Mike Shulman Jan 14 at 17:59
• See Sections 2.1 and 2.6 of the same book for what happens when the spaces are not compact. Simple maps are then characterized as hereditary weak equivalences (Theorem 2.1.7, due to Lacher), and maps that are locally trivial up to simple maps are Serre fibrations (Prop. 2.6.7). – John Rognes Jan 14 at 18:14
• Thanks! Could you add precise statements of those results to the answer? – Mike Shulman Jan 14 at 19:08