Theorem 1.4.6.1 (0079): Joyal

Theorem 1.4.6.1 (Joyal). Let $S_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$: The simplicial set $S_{\bullet }$ is an $\infty $-category.
$(2)$: The inclusion of simplicial sets $\Lambda ^2_1 \hookrightarrow \Delta ^2$ induces a trivial Kan fibration

\[ \operatorname{Fun}( \Delta ^2, S_{\bullet } ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, S_{\bullet } ). \]

Proof of Theorem 1.4.6.1. Let $S_{\bullet }$ be a simplicial set and let $p: \operatorname{Fun}( \Delta ^2, S_{\bullet } ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, S_{\bullet } )$ denote the restriction map. Then $p$ is a trivial Kan fibration if and only if every lifting problem

\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\partial \Delta }^ m \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^2, S_{\bullet }) \ar [d]^{p} \\ \Delta ^ m \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( \Lambda ^2_1, S_{\bullet }) } \]

admits a solution. Unwinding the definitions, we see that this is equivalent to the requirement that every lifting problem of the form

\[ \xymatrix@C =40pt@R=40pt{ (\Delta ^ m \times \Lambda ^2_1) \underset { \operatorname{\partial \Delta }^ m \times \Lambda ^2_1}{\coprod } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \ar [d]^{i} \ar [r] & S_{\bullet } \ar [d] \\ \Delta ^ m \times \Delta ^2 \ar [r] \ar@ {-->}[ur] & \Delta ^0 } \]

admits a solution. Let $T$ be the collection of all morphisms of simplicial sets which have the left lifting property with respect to the projection $S_{\bullet } \rightarrow \Delta ^0$. Then $p$ is a trivial Kan fibration if and only if $T$ contains each of the inclusion maps

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2. \]

Since $T$ is weakly saturated (Proposition 1.4.4.16), this is equivalent to the requirement that $T$ contains all inner anodyne morphisms (Lemma 1.4.6.9), which is in turn equivalent to the requirement that $S_{\bullet }$ is an $\infty $-category (Proposition 1.4.6.7). $\square$