Definition 1.1.0.4. Let $\operatorname{\mathcal{C}}$ be a category. A simplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. A cosimplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}\rightarrow \operatorname{\mathcal{C}}$.
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