Notation 3.5.2.8. Let $\operatorname{Lin}_{\neq \emptyset }$ denote the category whose objects are nonempty finite linearly ordered sets, and whose morphisms are nondecreasing functions. Note that, if $S$ is a finite subset of the unit interval $[0,1]$, then the complement $[0,1] \setminus S$ has finitely many connected components. Moreover, there is a unique linear ordering on the set $\pi _0( [0,1] \setminus S)$ for which the quotient map

\[ ( [0,1] \setminus S) \rightarrow \pi _0( [0,1] \setminus S) \]

is nondecreasing. We can therefore regard $\pi _0( [0,1] \setminus S)$ as an object of the category $\operatorname{Lin}_{\neq \emptyset }$.