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Variant 7.1.1.16. It will often be useful to extend the terminology of Definition 7.1.1.15, replacing the individual simplicial set $K$ by a collection of simplicial sets. For example:

  • We say that an $\infty $-category $\operatorname{\mathcal{C}}$ admits finite limits if, for every finite simplicial set $K$ (Definition 3.5.1.1), every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ admits a limit.

  • We say that an $\infty $-category $\operatorname{\mathcal{C}}$ admits finite colimits if, for every finite simplicial set $K$, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ admits a colimit.

  • We say that an $\infty $-category $\operatorname{\mathcal{C}}$ admits small limits if, for every (small) simplicial set $K$, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ admits a limit.

  • We say that an $\infty $-category $\operatorname{\mathcal{C}}$ admits small colimits if, for every (small) simplicial set $K$, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ admits a colimit.