Kerodon

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Proposition 7.4.5.1. Let $f: K \rightarrow \operatorname{\mathcal{S}}$ be a diagram. Then:

  • An extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is a limit diagram if and only if it is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.

  • An extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{S}}$ is a colimit diagram if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Proof. It follows immediately from the definitions that a diagram in $\operatorname{\mathcal{S}}$ which is a limit (or colimit) diagram in the larger $\infty $-category $\operatorname{\mathcal{QC}}$, then it is already a limit (or colimit) diagram in $\operatorname{\mathcal{S}}$ (see Variant 7.1.3.10). To prove the converse implications, we must show that the inclusion functor $\iota : \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{QC}}$ preserves all limits and colimits. This follows from Corollary 7.1.3.21, since the functor $\iota $ admits both left and right adjoints (Example 6.2.2.12). $\square$