$\newcommand{\O}{\mathcal{O}}$In the book: "Simplicial Homotopy Theory", by Goerss-Jardine, they 'prove' that every minimal fibration $q:X\to Y$ of simplicial sets $X,Y$ has $|q|:|X|\to|Y|$ a Serre fibration. However, their proof is very, very terse: I've spent a while thinking about the missing details, and I think I have a proper proof, but I'd really like commentary on how correct my interpretation is ('interpretation' in that I've had to guess what the proof idea is).
First of all I should note that this theorem is due to Gabriel-Zisman in their book: "Calculus of Fractions and Homotopy Theory", but I find their writing even more mysterious.
It will become important later to note that it suffices to work in the category $\mathsf{CGHaus}$ to demonstrate $|q|$ is a Serre fibration. Then, it will remain a Serre fibration even in $\mathsf{Top}$.
What Goerss and Jardine actually show:
It is enough to suppose $Y$ has only finitely many non-degenerate simplices, since any map $I^n\times I\to|Y|$ will fall into some finite subcomplex of $Y$ for compactness reasons, and similarly we may suppose $Y$ is connected. Let $F$ be the simplicial fibre of $q$ over some vertex $\ast$ of $Y$. Given a pushout diagram:
Where $\beta$ is a nondegenerate simplex of $Y$, i.e. $Y$ is the result of attaching $\beta$ to $Z$, and given any open set $U$ of $|Z|$ where there is a fibrewise homeomorphism: $$U\times_{|Y|}|X|\cong U\times|F|$$For 'fibrewise' in the sense that this homeomorphism preserves the projections to $U$, there is $U^1:=|\alpha|^{-1}(U)$ an open subset of $|\partial\Delta^n|$ and some open set $V^1\subseteq|\Delta^n|$ with $U^1$ a retract of $V^1$ and a fibrewise homeomorphism: $$(V^1\cup_{U^1}U)\times_{|Y|}|X|\cong(V^1\cup_{U^1}U)\times|F|$$
Where the proof of this last assertion uses the connectivity assumption on $Y$ but not the 'finitely many nondegenerate simplices' assumption. The set $V^1\cup_{U^1}U$ is the open subset of $|Y|$ defined in terms of the implicit pushout: really, from a point-set perspective, it is the set: $$|\beta|(V^1)\cup\iota(U)$$Where $\iota:|Z|\hookrightarrow|Y|$ is the (closed) subspace inclusion.
They remark at the beginning that the idea is to show $|q|$ is locally trivial. It turns out that any locally trivial map is a Serre fibration, where a map $f:A\to B$ is said to be locally trivial if for every $b\in B$ there is an open neighbourhood $\O_b$ of $b$ and a nonempty space $F$ such that: $$f^{-1}(\O_b)\cong\O_b\times F$$Which is fibrewise in the sense that the projections to $\O_b$ are preserved.
The following is my attempt at completing their proof.
The statement that they prove, and their method of proof, holds if $Y$ is replaced with any subcomplex $Z'$ of $Y$. $|Y|$ is obtained as a finite sequence of successive pushouts as in the above diagram, where $\beta$ and $n$ range over all nondegenerate simplices of $Y$ (except, $Y$ is replaced with $Z'$). This invites an inductive proof. We want to start with some basic subcomplex $Z_0$ of $Y$ for which we know $|q|$ is locally trivial over the geometric realisation $|Z_0|$, and somehow extend to $|q|$ being locally trivial over all of $|Y|$. Well, $|q|$ is trivially locally trivial over $Z_0=\emptyset\subseteq Y$.
Suppose, inductively, that we are given a subcomplex $Z$ of $Y$ with strictly fewer non-degenerate simplices, in such a way that there is some $n\in\Bbb N$ for which $Z_k=Y_k$ for all $k<n$ and $Z_n\subsetneq Y_n$, such that $|q|$ is locally trivial over $|Z|$ in the sense that $|q|_{|q|^{-1}|Z|}:|q|^{-1}(|Z|)\to|Z|$ is a locally trivial map. There is some $n\in\Bbb N$ and a nondegenerate $\beta\in Y_n$ that is not contained in $Z_n$. Let's call the pushout of: $$\Delta^n\hookleftarrow\partial\Delta^n\overset{\beta_{\partial\Delta^n}}{\longrightarrow}Z$$The subcomplex $Z'\subseteq Y$.
Since geometric realisation is cocontinuous, I know that the realisation of this is a topological pushout; $|Z'|$ is the result of an attaching square. I know that a subset of $|Z'|$ is open iff. its preimages in $|\Delta^n|$ and in $|Z|$ are both open. To check the local triviality criterion, take a $b\in|Z|$. To start with, take $b\in|Z|\cap|Z'|$. I know that there is a $|Z|$-open neighbourhood $U$ of $b$ that witnesses the local triviality of $|q|$ on $|Z|$. By what Goerss-Jardine proved, I know there is some $\O_b:=(V^1\cup_{U^1}U)$ a $|Z'|$-open neighbourhood of $b$ and a nonempty space $|F|$ that: $$|q|^{-1}(\O_b)\cong\O_b\times|F|$$Fibrewise. That's all that we need to show.
For $b\in|Z'|\setminus|Z|$, I can consider $\O_b:=|Z'|\setminus|Z|$ itself, which is open (and contains $b$) since it is known that $|Z|\hookrightarrow|Z'|$ is a closed subspace inclusion in this context. Let $F'$ be the simplicial fibre of $q$ over the vertex $\beta(0):\Delta^0\overset{0}{\hookrightarrow}\Delta^n\overset{\beta}{\longrightarrow}Z'$: by an earlier result of Goerss-Jardine, involving the minimality of $q$, we know: $$\beta^{-1}(q)\cong\Delta^n\times F'$$Fibrewise, in that the projections to $\Delta^n$ are preserved (the preimage of $q$ being, here, a pullback of the arrow $\Delta^n\overset{\beta}{\longrightarrow}Y$).
Taking realisations (in $\mathsf{CGHaus}$) (where $|.|$ preserves limits, in particular pullbacks such as $\beta^{-1}(q)$) I know that: $$|\beta|^{-1}(|q|)\cong|\Delta^n|\times|F'|$$Fibrewise, and I am fairly certain (I checked), since $|\beta|$ restricts to a homeomorphism on $|\Delta^n|\setminus|\partial\Delta^n|$ by pushout properties, that this induces: $$|q|^{-1}(\O_b)\cong\O_b\times|F'|$$
So $\O_b$ witnesses local triviality of $|q|$ at $b$. This will hold for any $b\in|Z|\setminus|Z'|$, so we are done.
Inducting upwards finitely many times, being sure to add all missing $n$-simplices before introducing new $(n+1)$-simplices, we arrive at $|q|$ being locally trivial (in $\mathsf{CGHaus}$) over all of $|Y|$, so that $|q|$ is a Serre fibration as required.
Does that make sense? Is there a better way?
I really appreciate any comments. This theorem has caused me some headaches!