Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

Variant 1.4.6.8. Let $S_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S_{\bullet }$ is isomorphic to the nerve of a category.

$(2)$

For every inner anodyne map of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$ and every map $f_0: A_{\bullet } \rightarrow S_{\bullet }$, there exists a unique map $f: B_{\bullet } \rightarrow S_{\bullet }$ such that $f_0 = f \circ i$.

Proof. Let us regard the simplicial set $S_{\bullet }$ as fixed, and let $T$ be the collection of all morphisms of simplicial sets $i: A_{\bullet } \rightarrow B_{\bullet }$ for which the induced map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( B_{\bullet }, S_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A_{\bullet }, S_{\bullet } )$ is bijective. Then $T$ is weakly saturated (in the sense of Definition 1.4.4.15). It follows that $(2)$ is equivalent to the following a priori weaker assertion:

$(2')$

For every pair of integers $0 < i < n$, the map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_ i, S_{\bullet } )$ is bijective.

The equivalence of $(1)$ and $(2')$ is the content of Proposition 1.2.3.1. $\square$