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. $\square$