The Stacks project

Lemma 14.32.3. Let $A$, $B$ be sets, and that $f : A \to B$ is a map. Consider the simplicial set $U$ with $n$-simplices

\[ A \times _ B A \times _ B \ldots \times _ B A\ (n + 1 \text{ factors)}. \]

see Example 14.3.5. If $f$ is surjective, the morphism $U \to B$ where $B$ indicates the constant simplicial set with value $B$ is a trivial Kan fibration.

Proof. Observe that $U$ fits into a cartesian square

\[ \xymatrix{ U \ar[d] \ar[r] & \text{cosk}_0(B) \ar[d] \\ B \ar[r] & \text{cosk}_0(A) } \]

Since the right vertical arrow is a trivial Kan fibration by Lemma 14.32.1, so is the left by Lemma 14.30.3. $\square$


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