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Definition (The Simplex Category). We define a category $\operatorname{{\bf \Delta }}$ as follows:

  • The objects of $\operatorname{{\bf \Delta }}$ are linearly ordered sets of the form $[n]$ for $n \geq 0$.

  • A morphism from $[m]$ to $[n]$ in the category $\operatorname{{\bf \Delta }}$ is a function $\alpha : [m] \rightarrow [n]$ which is nondecreasing: that is, for each $0 \leq i \leq j \leq m$, we have $0 \leq \alpha (i) \leq \alpha (j) \leq n$.

We will refer to $\operatorname{{\bf \Delta }}$ as the simplex category.