Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 5.1.4.13. Let $f: X \hookrightarrow Y$ be a monomorphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is left anodyne.

$(2)$

The morphism $f$ is a covariant equivalence relative to $Y$.

$(3)$

For every morphism of simplicial sets $Y \rightarrow S$, the morphism $f$ is a covariant equivalence relative to $S$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 5.1.3.6, and the implication $(2) \Rightarrow (3)$ from Remark 5.1.4.7. We will show that $(3)$ implies $(1)$. For this, we must show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r] & Z \ar [d]^{q} \\ Y \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution, provided that $q$ is a left fibration of simplicial sets (Corollary 4.2.2.9). To prove this, it will suffice to show that the natural map $\theta : \operatorname{Fun}_{/S}(Y,Z) \rightarrow \operatorname{Fun}_{/S}(X,Z)$ is surjective on vertices. In fact, our assumption that $f$ is a monomorphism and a covariant equivalence relative to $S$ guarantees that $\theta $ is a trivial Kan fibration of simplicial sets (Corollary 5.1.4.12). $\square$