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Proposition Let $f: X_{\bullet } \rightarrow S_{\bullet }$ be a morphism of simplicial sets. The following conditions are equivalent:


The morphism $f$ is a covering map.


For every map of standard simplices $u: \Delta ^ m \rightarrow \Delta ^ n$, composition with $u$ induces a bijection $X_{n} \rightarrow X_{m} \times _{ S_{m} } S_{n}$.


For every $n$-simplex $\sigma : \Delta ^ n \rightarrow S_{\bullet }$, the projection map $\Delta ^{n} \times _{S_{\bullet }} X_{\bullet } \rightarrow \Delta ^ n$ restricts to an isomorphism on each connected component of $\Delta ^ n \times _{S_{\bullet }} X_{\bullet }$.

Proof. Assume first that $(1)$ is satisfied; we will prove $(2)$. Let $u: \Delta ^ m \rightarrow \Delta ^ n$ be a morphism of simplicial sets. Choose a vertex $v: \Delta ^0 \rightarrow \Delta ^ m$. It follows from Example that $v$ and $u \circ v$ are anodyne morphisms of simplicial sets. Invoking Remark, we conclude that the right square and outer rectangle in the diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{n} \ar [d] \ar [r]^-{\circ u} & X_{m} \ar [r]^-{ \circ v} \ar [d] & X_{0} \ar [d] \\ S_{n} \ar [r]^-{ \circ u} & S_{m} \ar [r]^-{\circ v} & S_0 } \]

are pullback diagrams. It follows that the left square is a pullback diagram as well.

We next show that $(2)$ implies $(3)$. Fix a map $\sigma : \Delta ^ n \rightarrow S_{\bullet }$, and let $T = X_{n} \times _{S_{n}} \{ \sigma \} $ denote the collection of all $n$-simplices $\tau $ of $X_{\bullet }$ satisfying $f( \tau ) = \sigma $. To prove $(3)$, it will suffice to show that the tautological map

\[ g: \coprod _{\tau \in T} \Delta ^ n \rightarrow \Delta ^ n \times _{ S_{\bullet } } X_{\bullet } \]

is an isomorphism of simplicial sets. Equivalently, we must show that for every map of simplices $u: \Delta ^ m \rightarrow \Delta ^ n$, the induced map $T \rightarrow X_{m} \times _{S_{m}} \{ \sigma \circ u \} $ is bijective, which follows immediately from $(2)$.

We now complete the proof by showing that $(3)$ implies $(1)$. Assume that $(3)$ is satisfied. We wish to show that, for every pair of integers $0 \leq i \leq n$ with $n \geq 1$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar@ {^{(}->}[d] \ar [r] & X_{\bullet } \ar [d]^{f} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & S_{\bullet } } \]

admits a unique solution. To prove this, we are free to replace $f$ by the projection map $\Delta ^ n \times _{S_{\bullet }} X_{\bullet } \rightarrow \Delta ^ n$, and thereby reduce to the case where $S_{\bullet }$ is a standard simplex. In this case, assumption $(3)$ guarantees that each connected component of $X_{\bullet }$ is isomorphic to $S_{\bullet }$. The desired result now follows from the observation that the simplicial sets $\Lambda ^{n}_{i}$ and $\Delta ^ n$ are connected. $\square$