Kerodon

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Example 7.1.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • The morphism $f$ is an isomorphism from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Definition 1.3.6.1).

  • The morphism $f$ exhibits $X$ as a limit of the diagram $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$.

  • The morphism $f$ exhibits $Y$ as a colimit of the diagram $\{ X\} \hookrightarrow \operatorname{\mathcal{C}}$.