$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 6.3.6.13. Suppose we are given a commutative diagram of simplicial sets
6.15
\begin{equation} \begin{gathered}\label{equation:universal-localization-pushout} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [dr]^{ f_{01} } \ar [rr] \ar [dd] & & X_{0} \ar [dd] \ar [dr]^{ f_{0} } & \\ & S_{01} \ar [rr] \ar [dd] & & S_{0} \ar [dd] \\ X_{1} \ar [rr] \ar [dr]^{f_1} & & X \ar [dr]^{f} & \\ & S_{1} \ar [rr] & & S } \end{gathered} \end{equation}
with the following properties:
- $(a)$
The front and back faces
\[ \xymatrix@R =50pt@C=50pt{ S_{01} \ar [r] \ar [d] & S_0 \ar [d] & X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ S_1 \ar [r] & S & X_1 \ar [r] & X } \]
are pushout squares.
- $(b)$
The morphisms $S_{01} \rightarrow S_{0}$ and $X_{01} \rightarrow X_{0}$ are monomorphisms.
- $(c)$
The morphisms $f_{01}$, $f_{0}$, and $f_{1}$ are universally localizing.
Then the morphism $f$ is universally localizing.
Proof.
Fix a morphism of simplicial sets $T \rightarrow S$; we wish to show that the projection map $f_{T}: T \times _{S} X \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to some collection of morphisms $W$. Since the hypotheses of Proposition 6.3.6.13 are stable with respect to pullback, we may assume without loss of generality that $T = S$. Let $W_0$ be the collection of edges $w$ of $X_0$ having the property that $f_0(w)$ is a degenerate edge of $S_0$, and define $W_{1}$ and $W_{01}$ similarly. Combining assumption $(c)$ with Proposition 6.3.6.3, we conclude that the morphism $f_0$ (respectively $f_1$, $f_{01}$) exhibits the simplicial set $S_0$ (respectively $S_1$, $S_{01}$) as a localization of $X_{0}$ (respectively $X_1$, $X_{01}$) with respect to $W_0$ (respectively $W_{1}$, $W_{01}$). Let $W$ be the collection of edges of $X$ given by the union of the images of $W_0$ and $W_1$. Note that conditions $(a)$ and $(b)$ guarantee that the front and back faces of the diagram (6.15) are categorical pushout squares (Proposition 4.5.4.11). Applying Proposition 6.3.4.2, we conclude that $f$ exhibits $S$ as a localization of $X$ with respect to $W$.
$\square$