Kerodon

Remark 1.5.2.13. In the situation of Convention 1.5.2.12, suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$, where $\operatorname{\mathcal{C}}_0$ is an ordinary category. Then giving a commutative diagram in the $\infty $-category $\operatorname{\mathcal{C}}$ (in the sense of Convention 1.5.2.12) is equivalent to giving a commutative diagram in the ordinary category $\operatorname{\mathcal{C}}_0$ (in the sense of Definition 1.5.2.5). In this case, commutativity is a property that the underlying diagram (indexed by a $1$-dimensional simplicial set) does or does not possess. For a general $\infty $-category $\operatorname{\mathcal{C}}$, commutativity of a diagram in $\operatorname{\mathcal{C}}$ is not a property but a structure; to promote a diagram to a commutative diagram, one must specify additional data to witness the requisite commutativity.