We now study a special class of inner fibrations.
Definition 4.1.5.1. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We say that $f$ is an inner covering map if, for every pair of integers $0 < i < n$, every lifting problem has a unique solution.
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{f} \\ \Delta ^ n \ar [r] & S } \]
Example 4.1.5.2. Every covering map of simplicial sets (in the sense of Definition 3.1.4.1) is an inner covering map. In particular, if $f: X \rightarrow S$ is a covering map of topological spaces, then the induced map $\operatorname{Sing}_{\bullet }(f): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is an inner covering of simplicial sets (Proposition 3.1.4.9).
Example 4.1.5.3. Let $X$ be a simplicial set. Then the projection map $f: X \rightarrow \Delta ^0$ is an inner covering map if and only if $X$ is isomorphic to the nerve of a category (this is a restatement of Proposition 1.3.4.1).
Remark 4.1.5.4. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Then $f$ is an inner covering map if and only if the opposite morphism $f^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ is an inner covering map.
Remark 4.1.5.5. 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 an inner covering map if and only if both $f$ and $\delta $ are inner fibrations. In particular, every inner covering map is an inner fibration.
Example 4.1.5.6. Let $f: X \hookrightarrow S$ be a monomorphism of simplicial sets, so that the relative diagonal $\delta : X \hookrightarrow X \times _{S} X$ is an isomorphism. Then $f$ is an inner fibration if and only if it is an inner covering. In particular, if $\operatorname{\mathcal{C}}$ is an $\infty $-category and $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is subcategory, then the inclusion map $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is an inner covering.
Remark 4.1.5.7. Suppose we are given a pullback diagram of simplicial sets If $f$ is an inner covering map, then $f'$ is also an inner covering map.
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^-{f'} & X \ar [d]^-{f} \\ S' \ar [r] & S. } \]
Remark 4.1.5.8. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of simplicial sets. Suppose that $g$ is an inner covering map. Then $f$ is an inner covering map if and only if $g \circ f$ is an inner covering map. In particular, the collection of inner covering maps is closed under composition.
Remark 4.1.5.9. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $f$ is an inner covering map (Definition 4.1.5.1).
For every square diagram of simplicial sets
where $i$ is inner anodyne, there exists a unique dotted arrow rendering the diagram commutative.
\[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]^{i} \ar [r] & X_{} \ar [d]^{f} \\ B_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]
Proposition 4.1.5.10. Let $\operatorname{\mathcal{C}}$ be a category, and let $f: X \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. Then $f$ is an inner covering map if and only if $X$ is isomorphic to the nerve of a category.
Proof. Combine Remark 4.1.5.8 with Example 4.1.5.2. $\square$
Corollary 4.1.5.11. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Then $f$ is an inner covering if and only if, for every simplex $\sigma : \Delta ^ n \rightarrow S$, the fiber product $\Delta ^{n} \times _{S} X$ is isomorphic to the nerve of a category.
Proof. Suppose $f$ is an inner covering. For every simplex $\sigma : \Delta ^ n \rightarrow S$, it follows from Remark 4.1.5.7 that the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is also an inner covering map, so that $\Delta ^ n \times _{S} X$ is isomorphic to the nerve of a category by virtue of Proposition 4.1.5.10. Conversely, to show that $f$ is an inner covering map, it will suffice to show that every lifting problem
has a unique solution for $0 < i < n$. If the fiber product $\Delta ^{n} \times _{S} X$ is the nerve of a category, then the existence and uniqueness of the desired solution follow from (and uniqueness) of the desired solution follow from Proposition 1.3.4.1. $\square$
Exercise 4.1.5.12. Let $f: X \rightarrow S$ be an inner covering map of simplicial sets and let $i: A \hookrightarrow B$ be any monomorphism of simplicial sets. Show that the restriction map is also an inner covering map. If $i$ is inner anodyne, show that $\theta $ is an isomorphism.
\[ \theta : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]