Kerodon

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

Corollary 4.2.3.18. Let $f: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is a left covering map of simplicial sets.

$(2)$

For every category $\operatorname{\mathcal{C}}$ and every morphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow S$, the pullback $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{S} X$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}$, and $f$ induces a left covering functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.

$(3)$

For every $n$-simplex $\Delta ^ n \rightarrow \S $, the fiber product $\Delta ^ n \times _{S} X$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}$ and the induced map $\operatorname{\mathcal{E}}\rightarrow [n]$ is a left covering functor.