Kerodon

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Definition 1.4.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A diagram in $\operatorname{\mathcal{C}}$ is a map of simplicial sets $f: K_{\bullet } \rightarrow \operatorname{\mathcal{C}}$. We will also refer to a map $f: K_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ as a diagram in $\operatorname{\mathcal{C}}$ indexed by $K_{\bullet }$, or a $K_{\bullet }$-indexed diagram in $\operatorname{\mathcal{C}}$.

If $\operatorname{\mathcal{C}}$ is an ordinary category, then a ($K_{\bullet }$-indexed) diagram in $\operatorname{\mathcal{C}}$ is a ($K_{\bullet }$-indexed) diagram in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

In the special case where $K_{\bullet }$ is the nerve $\operatorname{N}_{\bullet }(I)$ of a partially ordered set $I$ (Remark 1.2.1.8), we will refer to a map $f: K_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ as a diagram in $\operatorname{\mathcal{C}}$ indexed by $I$, or an $I$-indexed diagram in $\operatorname{\mathcal{C}}$.