Kerodon

Proof of Proposition 3.6.6.5. Assume first that the map $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a Kan fibration of simplicial sets. It follows that, for each $n \geq 0$, $\operatorname{Sing}_{\bullet }(q)$ is weakly right orthogonal to the inclusion map $\{ 0\} \times \Delta ^ n \hookrightarrow \Delta ^1 \times \Delta ^ n$ (which is anodyne, by virtue of Proposition 3.1.2.9). It follows that the continuous function $q$ is weakly right orthogonal to the map of geometric realizations $| \{ 0\} \times \Delta ^ n | \hookrightarrow | \Delta ^1 \times \Delta ^ n |$, which can be identified with the inclusion $\{ 0\} \times | \Delta ^ n | \hookrightarrow [0,1] \times | \Delta ^ n |$ (see Corollary 3.6.2.2). Allowing $n$ to vary, we deduce that $q$ is a Serre fibration.

We now prove the converse. Suppose that $q$ is a Serre fibration; we wish to show that the induced map of simplicial sets $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is weakly right orthogonal to the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for every pair of integers $0 \leq i \leq n$ with $n > 0$. Equivalently, we wish to show that $q$ is weakly right orthogonal to the inclusion of geometric realizations $\iota : | \Lambda ^{n}_{i} | \hookrightarrow | \Delta ^ n |$. We proceed by refining the proof of Proposition 1.2.5.8. Define a continuous function $c: | \Delta ^{n} | \rightarrow [0,1]$ by the formula $c( t_0, t_1, \cdots , t_ n) = \min \{ t_0, \ldots , t_{i-1}, t_{i+1}, \ldots , t_ n \} $. Let $h: [0,1] \times | \Delta ^ n | \rightarrow | \Delta ^ n |$ be the continuous function given by the formula

By construction, the composition

is the identity map. Moreover, the function $(c,\operatorname{id})$ carries the horn $| \Lambda ^{n}_{i} | \subset | \Delta ^ n |$ to the closed subset $\{ 0\} \times | \Delta ^ n | \subseteq | \Delta ^{n} |$, and the function $h$ carries $\{ 0\} \times | \Delta ^ n |$ to the horn $| \Lambda ^{n}_{i} | \subset | \Delta ^ n |$. It follows that $h$ and $(c,\operatorname{id})$ exhibit $\iota $ as a retract of the inclusion map $\iota ': \{ 0\} \times | \Delta ^ n | \hookrightarrow [0,1] \times | \Delta ^ n |$ in the category of topological spaces. Consequently, to show that $q$ is weakly right orthogonal to $\iota $, it will suffice to show that it is weakly right orthogonal to $\iota '$ (Proposition 1.5.4.9), which follows immediately from our assumption that $q$ is a Serre fibration. $\square$

Proof. The implication $(3) \Rightarrow (2) \Rightarrow (1)$ are immediate from the definition. We will complete the proof by showing that $(1)$ implies $(3)$. Using Corollary 3.6.2.2, we observe that every lifting problem of the form (3.83) can be rewritten as a lifting problem

in the category of simplicial sets. If $q$ is Serre fibration, then $\operatorname{Sing}_{\bullet }(q)$ is a Kan fibration (Proposition 3.6.6.5), so the existence of the desired lifting follows from the observation that $\iota $ is an anodyne morphism (Proposition 3.1.2.9). $\square$

Proof. Let $\operatorname{\mathcal{U}}$ be the collection of all open subsets $U \subseteq S$ for which the map $q_{U}$ is a Serre fibration. Suppose we are given a finite simplicial set $B$ and a simplicial subset $A \subseteq B$. We will say that a lifting problem

is $\operatorname{\mathcal{U}}$-small if, for every element $s \in [0,1]$ and every simplex $\sigma : \Delta ^{k} \rightarrow B$, the image of the composite map

is contained in some open set belonging to the cover $\operatorname{\mathcal{U}}$. We first claim that every $\operatorname{\mathcal{U}}$-small lifting problem admits a solution. Proceeding by induction on the number of simplices of $B$ which do not belong to $A$, we can reduce to the case where $B$ is a standard simplex and $A$ is its boundary. In this case, it follows from our $\operatorname{\mathcal{U}}$-smallness assumption and the compactness of the product $[0,1] \times | B |$ that there exists some integer $m \gg 0$ with the property that, for each $1 \leq k \leq m$, the composite map

has image contained in some open set $U_{k} \in \operatorname{\mathcal{U}}$. Writing $\iota $ as a composition of inclusion maps

we are reduced to solving a finite sequence of lifting problems

which is possible by virtue of our assumption that $q_{U_ k}$ is a Serre fibration (Corollary 3.6.6.8).

Fix an integer $n \geq 0$; we wish to show that every lifting problem

admits a solution. Fix an integer $t \geq 0$, and $B = \operatorname{Sd}^{t}( \Delta ^{n} )$ denote the $t$-fold subdivision of $\Delta ^ n$. Then Proposition 3.3.3.6 supplies a homeomorphism $| B | \simeq | \Delta ^ n |$, which we can use to rewrite (3.84) as a lifting problem

It follows from Lemma 3.4.6.7 that the lifting problem (3.85) is $\operatorname{\mathcal{U}}$-small for $t \gg 0$, and therefore admits a solution by the first step of the proof. $\square$

Proof. By virtue of Proposition 3.6.6.10, it suffices to check this locally on $S$ and we may therefore assume that there exists a pullback diagram

for some topological space $Y$. Using Remark 3.6.6.3, we are reduced to showing that the projection map $Y \rightarrow \{ \ast \} $ is a Serre fibration, which follows from Example 3.6.6.2. $\square$