Definition 10.2.1.12 (Augmented Simplicial Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An augmented simplicial object of $\operatorname{\mathcal{C}}$ is a functor from the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} )$ to $\operatorname{\mathcal{C}}$. An augmented cosimplicial object is a functor from the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+} )$ to $\operatorname{\mathcal{C}}$.
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