# Kerodon

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Definition 7.1.3.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty$-categories and let $K$ be a simplicial set. We will say that the functor $F$ creates $K$-indexed limits if the following condition is satisfied:

• Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ admits a limit in $\operatorname{\mathcal{D}}$. Then $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

We say that the functor $F$ creates $K$-indexed colimits if it satisfies the following dual condition:

• Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ admits a colimit in $\operatorname{\mathcal{D}}$. Then $u$ can be extended to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.