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Notation 2.4.3.1 (Simplicial Path Categories). Let $(Q, \leq )$ be a partially ordered set, and let $\operatorname{Path}_{(2)}[Q]$ denote the path $2$-category of $Q$ (Construction 2.3.6.1). We let $\operatorname{Path}[Q]_{\bullet }$ denote the simplicial category obtained from the strict $2$-category $\operatorname{Path}_{(2)}[Q]$ by applying the construction of Example 2.4.2.7. More concretely, we can describe the simplicial category $\operatorname{Path}[Q]_{\bullet }$ as follows:

  • The objects of $\operatorname{Path}[Q]_{\bullet }$ are the elements of the partially ordered set $Q$.

  • If $x$ and $y$ are elements of $Q = \operatorname{Ob}(\operatorname{Path}[Q]_{\bullet } )$, then $\operatorname{Hom}_{\operatorname{Path}[Q]}( x, y)_{\bullet }$ is the nerve of the partially ordered set of finite linearly ordered subsets $\{ x = x_0 < x_1 < \cdots < x_ m = y \} \subseteq Q$ with least element $x$ and largest element $y$, ordered by reverse inclusion.

  • For each element $x \in Q = \operatorname{Ob}(\operatorname{Path}[Q]_{\bullet } )$, the identity morphism $\operatorname{id}_{x}$ is the singleton $\{ x\} \in \operatorname{Hom}_{\operatorname{Path}[Q]}(x,x)_0$.

  • For $x,y,z \in Q = \operatorname{Ob}(\operatorname{Path}[Q] )$, the composition law

    \[ \operatorname{Hom}_{\operatorname{Path}[Q]}(y,z)_{\bullet } \times \operatorname{Hom}_{ \operatorname{Path}[Q]}( x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Path}[Q]}(x,z)_{\bullet } \]

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

In the special case where $Q = [n] = \{ 0 < 1 < \cdots < n \} $, we denote the simplicial category $\operatorname{Path}[Q]_{\bullet }$ by $\operatorname{Path}[n]_{\bullet }$.