The Stacks project

Definition 14.3.1. Let $\mathcal{C}$ be a category.

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

   \[ U : \Delta ^{opp} \longrightarrow \mathcal{C} \]

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

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

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

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