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Construction (The Path $2$-Category of a Partially Ordered Set). Let $(Q, \leq )$ be a partially ordered set. We define a strict $2$-category $\operatorname{Path}_{(2)}[Q]$ as follows:

  • The objects of $\operatorname{Path}_{(2)}[Q]$ are the elements of $Q$.

  • Given elements $x,y \in Q$, we let $\underline{\operatorname{Hom}}_{ \operatorname{Path}[Q]_{(2)} }(x, y)$ denote the partially ordered set of all finite linearly ordered subsets

    \[ S = \{ x = x_0 < x_1 < \cdots < x_ n = y \} \subseteq Q \]

    having least element $x$ and greatest element $y$, ordered by reverse inclusion. We regard the partially ordered set $\underline{\operatorname{Hom}}_{\operatorname{Path}[Q]_{(2)}}(x,y)$ as a category, having a unique morphism $S \Rightarrow T$ when $T$ is contained in $S$.

  • For every element $x \in Q$, the identity $1$-morphism $\operatorname{id}_{x} \in \underline{\operatorname{Hom}}_{\operatorname{Path}[Q]}( x, x)$ is given by the singleton $\{ x\} $ (regarded as a linearly ordered subset of $Q$, having greatest and least element $x$).

  • For every triple of objects $x,y,z \in Q$, the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{Path}_{(2)}[Q]}(y,z) \times \underline{\operatorname{Hom}}_{\operatorname{Path}_{(2)}[Q]}(x,y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{Path}_{(2)}[Q]}(x,z) \]

    is given on objects by the construction $(S, T) \mapsto S \cup T$.

‘ We will refer to $\operatorname{Path}_{(2)}[Q]$ as the path $2$-category of $Q$.