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