The Stacks project

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

1. A cosimplicial object $U$ of $\mathcal{C}$ is a covariant functor $U$ from $\Delta$ to $\mathcal{C}$, in a formula:

   $$U : \Delta \longrightarrow \mathcal{C}$$

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

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

4. A morphism of cosimplicial objects $U \to U'$ is a transformation of functors.

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