Definition 14.5.1. Let $\mathcal{C}$ be a category.

A

*cosimplicial object $U$ of $\mathcal{C}$*is a covariant functor $U$ from $\Delta $ to $\mathcal{C}$, in a formula:\[ U : \Delta \longrightarrow \mathcal{C} \]If $\mathcal{C}$ is the category of sets, then we call $U$ a

*cosimplicial set*.If $\mathcal{C}$ is the category of abelian groups, then we call $U$ a

*cosimplicial abelian group*.A

*morphism of cosimplicial objects $U \to U'$*is a transformation of functors.The

*category of cosimplicial objects of $\mathcal{C}$*is denoted $\text{CoSimp}(\mathcal{C})$.

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