Definition 5.4.1.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We will say that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category if it satisfies the following axioms:
- $(1)$
Every morphism of simplicial sets $\Lambda ^{2}_{1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
- $(2)$
Every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ is thin.
- $(3)$
Let $n \geq 3$ and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0< 1 < n\} ) }$ is left-degenerate (see Example 1.1.2.8) Then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.
- $(4)$
Let $n \geq 3$ and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0< n-1 < n\} ) }$ is right-degenerate. Then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.