Definition 10.1.1.10 (The Augmented Simplex Category). For each integer $n \geq -1$, let $[n]$ denote the linearly ordered set $\{ 0 < 1 < \cdots < n \} $, so that $[-1]$ is the empty set. We let $\operatorname{{\bf \Delta }}_{+}$ denote the category whose objects are the linearly ordered sets $\{ [n] \} _{n \geq -1}$, and whose morphisms are nondecreasing functions. We will refer to $\operatorname{{\bf \Delta }}_{+}$ as the *augmented simplex category*.

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