Warning 2.4.4.5. We have now introduced several different notions of path category:
- $(a)$
To every directed graph $G$, Construction 1.3.7.1 associates an ordinary category $\operatorname{Path}[G]$.
- $(b)$
To every partially ordered set $Q$, Notation 2.4.3.1 associates a simplicial category $\operatorname{Path}[Q]_{\bullet }$.
- $(c)$
To every simplicial set $S_{\bullet }$, Proposition 2.4.4.3 associates a simplicial category $\operatorname{Path}[ S ]_{\bullet }$.
We will show below that these constructions are closely related:
- $(1)$
If $G$ is a directed graph and $S_{\bullet }$ denotes the associated simplicial set of dimension $\leq 1$ (Proposition 1.1.6.9), then the simplicial category $\operatorname{Path}[ S ]_{\bullet }$ of $(c)$ is constant, associated to the ordinary category $\operatorname{Path}[G]$ of $(a)$ (Proposition 2.4.4.7).
- $(2)$
If $Q$ is a partially ordered set, then the simplicial category $\operatorname{Path}[Q]_{\bullet }$ of $(b)$ can be identified with the simplicial category $\operatorname{Path}[ \operatorname{N}(Q) ]_{\bullet }$ of $(c)$, where $\operatorname{N}_{\bullet }(Q)$ denotes the nerve of $Q$ (Proposition 2.4.4.15).
- $(3)$
For any simplicial set $S_{\bullet }$, the simplicial category $\operatorname{Path}[ S ]_{\bullet }$ of $(c)$ has an underlying ordinary category $\operatorname{Path}[ S ]_0$, which can be described as the category $\operatorname{Path}[G]$ associated by $(a)$ to the underlying directed graph $G = \mathrm{Gr}( S_{\bullet } )$ of $S_{\bullet }$ (Proposition 2.4.4.13).
Assertions $(1)$ and $(2)$ imply that the path category constructions of §1.3.7 and §2.4.3 can be regarded as special cases of the construction $S_{\bullet } \mapsto \operatorname{Path}[ S ]_{\bullet }$. Assertion $(3)$ is a partial converse, which guarantees that the simplicial path category $\operatorname{Path}[S]_{\bullet }$ can be regarded as a simplicially enriched version of the classical path category studied in §1.3.7.