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 $\text{Simp}(\mathcal{C})$.

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