The Stacks project

Lemma 14.32.1. Let $f : V \to U$ be a morphism of simplicial sets. Let $n \geq 0$ be an integer. Assume

  1. The map $f_ i : V_ i \to U_ i$ is a bijection for $i < n$.

  2. The map $f_ n : V_ n \to U_ n$ is a surjection.

  3. The canonical morphism $U \to \text{cosk}_ n \text{sk}_ n U$ is an isomorphism.

  4. The canonical morphism $V \to \text{cosk}_ n \text{sk}_ n V$ is an isomorphism.

Then $f$ is a trivial Kan fibration.

Proof. Consider a solid diagram

\[ \xymatrix{ \partial \Delta [k] \ar[r] \ar[d] & V \ar[d] \\ \Delta [k] \ar[r] \ar@{-->}[ru] & U } \]

as in Definition 14.30.1. Let $x \in U_ k$ be the $k$-simplex corresponding to the lower horizontal arrow. If $k \leq n$ then the dotted arrow is the one corresponding to a lift $y \in V_ k$ of $x$; the diagram will commute as the other nondegenerate simplices of $\Delta [k]$ are in degrees $< k$ where $f$ is an isomorphism. If $k > n$, then by conditions (3) and (4) we have (using adjointness of skeleton and coskeleton functors)

\[ \mathop{Mor}\nolimits (\Delta [k], U) = \mathop{Mor}\nolimits (\text{sk}_ n\Delta [k], \text{sk}_ nU) = \mathop{Mor}\nolimits (\text{sk}_ n\partial \Delta [k], \text{sk}_ nU) = \mathop{Mor}\nolimits (\partial \Delta [k], U) \]

and similarly for $V$ because $\text{sk}_ n\Delta [k] = \text{sk}_ n\partial \Delta [k]$ for $k > n$. Thus we obtain a unique dotted arrow fitting into the diagram in this case also. $\square$


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