Kerodon

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

Corollary 5.6.1.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then $U$ is a left covering map (in the sense of Definition 4.2.3.8) if and only if the following pair of conditions is satisfied:

$(1)$

The induced map $\mathrm{h} \mathit{U}: \mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a left covering functor (in the sense of Definition 4.2.3.1).

$(2)$

The diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{E}}} ) \ar [d]^{\operatorname{N}_{\bullet }( \mathrm{h} \mathit{U} )} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}) } \]

is a pullback square.

Proof. The sufficiency of conditions $(1)$ and $(2)$ follows from Proposition 4.2.3.16 and Remark 4.2.3.15. To prove the converse, assume that $U$ is a left covering map. By virtue of Corollary 5.6.1.7, we may assume that $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for some morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$. Let us abuse notation by identifying $\mathscr {F}$ with a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets. Using Proposition 5.5.5.2, we can identify $\mathrm{h} \mathit{\operatorname{\mathcal{E}}}$ with the category of elements $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathscr {F}$ of Construction 5.5.1.1. Condition $(1)$ now follows from Remark 5.5.1.9, and condition $(2)$ by combining Example 5.5.4.8 with Remark 5.5.4.17. $\square$