Let $X$ and $S$ be topological spaces. Recall that a continuous function $f: X \rightarrow S$ is a covering map if every point $s \in S$ has an open neighborhood $U \subseteq S$ for which the inverse image $f^{-1}(U)$ is homeomorphic to a disjoint union of copies of $U$. This definition has a counterpart in the setting of simplicial sets:
Definition 3.1.4.1. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We say that $f$ is a covering map if, for every pair of integers $0 \leq i \leq n$ with $n > 0$, every lifting problem has a unique solution.
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{f} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & S } \]
Remark 3.1.4.2. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Then $f$ is a covering map if and only if the opposite morphism $f^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ is a covering map.
Remark 3.1.4.3. Let $f: X \rightarrow S$ be a morphism of simplicial sets, and let $\delta : X \rightarrow X \times _{S} X$ be the relative diagonal of $f$. Then $f$ is a covering map if and only if both $f$ and $\delta $ are Kan fibrations. In particular, every covering map is a Kan fibration.
Remark 3.1.4.4. Suppose we are given a pullback diagram of simplicial sets If $f$ is a covering map, then $f'$ is also a covering map.
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^-{f'} & X \ar [d]^-{f} \\ S' \ar [r] & S. } \]
Remark 3.1.4.5. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of simplicial sets. Suppose that $g$ is a covering map. Then $f$ is a covering map if and only if $g \circ f$ is a covering map. In particular, the collection of covering maps is closed under composition.
Remark 3.1.4.6. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $f$ is a covering map (Definition 3.1.4.1).
For every square diagram of simplicial sets
where $i$ is anodyne, there exists a unique dotted arrow rendering the diagram commutative.
\[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]^{i} \ar [r] & X_{} \ar [d]^{f} \\ B_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]
This follows by combining Remarks 3.1.2.7 and 3.1.4.3.
Proposition 3.1.4.7. Let $f: X_{} \rightarrow S_{}$ be a covering map of simplicial sets, and let $i: A_{} \hookrightarrow B_{}$ be any monomorphism of simplicial sets. Then the induced map is a covering map.
\[ \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]
Proof. By virtue of Remark 3.1.4.6, it will suffice to show that if $i': A' \hookrightarrow B'$ is an anodyne morphism of simplicial sets, then every lifting problem of the form
admits a unique solution. Equivalently, we must show that every lifting problem
admits a unique solution. This follows from Remark 3.1.4.6, since the left vertical map is anodyne (Proposition 3.1.2.9) and $f$ is a covering map. $\square$
Corollary 3.1.4.8. Let $f: X_{} \rightarrow S_{}$ be a covering map of simplicial sets. Then, for every simplicial set $B_{}$, composition with $f$ induces a covering map $\operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} )$.
Proposition 3.1.4.9. Let $f: X \rightarrow S$ be a covering map of topological spaces. Then the induced map $\operatorname{Sing}_{\bullet }(f): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a covering map of simplicial sets (in the sense of Definition 3.1.4.1).
Proof. Let $\delta : X \rightarrow X \times _{S} X$ be the relative diagonal of $f$. We first claim $\delta $ exhibits $X$ as a summand of $X \times _{S} X$ in the category of topological spaces (that is, it is a homeomorphism of $X$ onto a closed and open subset of the fiber product $X \times _{S} X$). To verify this, we can work locally on $S$ and thereby reduce to the case where $X$ is a product of $S$ with a discrete topological space, in which case the result is clear. It follows that the induced map of singular simplicial sets
is also the inclusion of a summand (Remark 1.2.2.4), and is therefore a Kan fibration by virtue of Example 3.1.1.4. Consequently, to show that $\operatorname{Sing}_{\bullet }(f)$ is a covering map, it will suffice to show that it is a Kan fibration (Remark 3.1.4.3). This is a special case of Corollary 3.6.6.11, since $f: X \rightarrow S$ exhibits $X$ as a fiber bundle over $S$ (with discrete fibers). $\square$
Warning 3.1.4.10. The converse of Proposition 3.1.4.9 is false. For example, let $f: X \rightarrow S$ be a continuous function between topological spaces where $S = \ast $ consists of a single point. In this case, the function $f$ is a covering map if and only if the topology on $X$ is discrete. However, the induced map of simplicial sets $\operatorname{Sing}_{\bullet }(f): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a covering map if and only if the simplicial set $\operatorname{Sing}_{\bullet }(X)$ is discrete: that is, if and only if every continuous function $[0,1] \rightarrow X$ is constant (Example 3.1.4.13). Many non-discrete topological spaces satisfy this weaker condition (for example, we could take $X$ to be the Cantor set).
Remark 3.1.4.11. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Then $f$ is a covering map (in the sense of Definition 3.1.4.1) if and only if the induced map of geometric realizations $| X | \rightarrow |S|$ is a covering map of topological spaces (see Proposition 
).
Covering maps of simplicial sets have a very simple local structure:
Proposition 3.1.4.12. 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 3.1.2.5 that $v$ and $u \circ v$ are anodyne morphisms of simplicial sets. Invoking Remark 3.1.4.6, we conclude that the right square and outer rectangle in the diagram
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
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
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. $\square$
Example 3.1.4.13. Let $X$ be a simplicial set. Then the unique morphism $f: X \rightarrow \Delta ^{0}$ is a covering map of simplicial sets if and only if $X$ is discrete (see Definition 1.1.5.10).
Corollary 3.1.4.14. Let $f: X \rightarrow S$ be a monomorphism of simplicial sets. The following conditions are equivalent:
The morphism $f$ exhibits $X$ as a summand of $S$ (Definition 1.2.1.1).
The morphism $f$ is a covering map.
The morphism $f$ is a Kan fibration.
Proof. The implication $(1) \Rightarrow (2)$ and $(2) \Rightarrow (3)$ are immediate. Moreover, if $f$ is a monomorphism, then the relative diagonal $\delta : X \rightarrow X \times _{S} X$ is an isomorphism, so the implication $(3) \Rightarrow (2)$ follows from Remark 3.1.4.3. We will complete the proof by showing that $(2) \Rightarrow (1)$. Let $u: \Delta ^{m} \rightarrow \Delta ^{n}$ be a morphism of standard simplices and let $\sigma : \Delta ^{n} \rightarrow S$ be a simplex of $S$; we wish to show that $\sigma $ factors through $f$ if and only if $\sigma \circ u$ factors through $f$. This follows immediately from the criterion of Proposition 3.1.4.12. $\square$