$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.2.4.12. Let $q: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $q$ is a left covering map, in the sense of Definition 4.2.3.8.
- $(2)$
Every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
admits a unique solution, provided that the morphism $i$ is inner anodyne.
- $(3)$
For every integer $n \geq 0$, the diagram of sets
\[ \xymatrix { \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, X ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \{ 0\} , X ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \{ 0\} , S) } \]
is a pullback square.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.2.4.5 and Remark 4.2.3.11, and the implication $(2) \Rightarrow (3)$ follows from Corollary 4.2.4.11. We will complete the proof by showing that $(3)$ implies $(1)$. Assume that condition $(3)$ is satisfied; we wish to show that, for $0 \leq j < n$, the left half of the diagram
\[ \xymatrix { \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, X) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_{j}, X) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \{ 0\} , X) \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_{j}, S) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \{ 0\} , S) } \]
is a pullback square. We proceed by induction on $n$. Assumption $(3)$ guarantees that the outer rectangle is a pullback, so we are reduced to showing that the square on the right is a pullback. This follows by combining our inductive hypothesis with Proposition 4.2.4.10.
$\square$