Notation 10.2.6.1. We define a category $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ as follows:
The objects of $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ are linearly ordered sets $[n] = \{ 0 < 1 < \cdots < n \} $, where $n$ is a nonnegative integer.
A morphism from $[m]$ to $[n]$ in the category $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ is a nondecreasing function $\alpha : [m] \rightarrow [n]$ satisfying $\alpha (0) = 0$.