Remark 1.2.2.11. The cosimplicial space $| \Delta ^{\bullet } |$ of Example 1.2.2.9 can be described more informally as follows:
To each nonempty finite linearly ordered set $I$, it assigns a topological simplex $| \Delta ^{I} |$ whose vertices are the elements of $I$: that is, the convex hull of the set $I$ inside the real vector space $\operatorname{\mathbf{R}}[I]$ generated by $I$.
To every nondecreasing map $\alpha : I \rightarrow J$, the induced map $| \Delta ^{I} | \rightarrow | \Delta ^{J} |$ is given by the restriction of the $\operatorname{\mathbf{R}}$-linear map $\operatorname{\mathbf{R}}[I] \rightarrow \operatorname{\mathbf{R}}[J]$ determined by $\alpha $. Equivalently, it is the unique affine map which coincides with $\alpha $ on the vertices of the simplex $| \Delta ^{I} |$.