The Stacks project

Definition 14.20.1. Let $\mathcal{C}$ be a category. Let $U$ be a simplicial object of $\mathcal{C}$. An augmentation $\epsilon : U \to X$ of $U$ towards an object $X$ of $\mathcal{C}$ is a morphism from $U$ into the constant simplicial object $X$.

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View Definition 14.20.1 as pdf