The Stacks project

Lemma 14.30.2. Let $f : X \to Y$ be a trivial Kan fibration of simplicial sets. For any solid commutative diagram

\[ \xymatrix{ Z \ar[r]_ b \ar[d] & X \ar[d] \\ W \ar[r]^ a \ar@{-->}[ru] & Y } \]

of simplicial sets with $Z \to W$ (termwise) injective a dotted arrow exists making the diagram commute.

Proof. Suppose that $Z \not= W$. Let $n$ be the smallest integer such that $Z_ n \not= W_ n$. Let $x \in W_ n$, $x \not\in Z_ n$. Denote $Z' \subset W$ the simplicial subset containing $Z$, $x$, and all degeneracies of $x$. Let $\varphi : \Delta [n] \to Z'$ be the morphism corresponding to $x$ (Lemma 14.11.3). Then $\varphi |_{\partial \Delta [n]}$ maps into $Z$ as all the nondegenerate simplices of $\partial \Delta [n]$ end up in $Z$. By assumption we can extend $b \circ \varphi |_{\partial \Delta [n]}$ to $\beta : \Delta [n] \to X$. By Lemma 14.21.7 the simplicial set $Z'$ is the pushout of $\Delta [n]$ and $Z$ along $\partial \Delta [n]$. Hence $b$ and $\beta $ define a morphism $b' : Z' \to X$. In other words, we have extended the morphism $b$ to a bigger simplicial subset of $Z$.

The proof is finished by an application of Zorn's lemma (omitted). $\square$

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