Kerodon

Remark 8.5.9.10. Using the universal property of the path category $\operatorname{\mathcal{C}}= \operatorname{Path}[ \operatorname{N}_{\leq 2}(\operatorname{Idem}) ]_{\bullet }$, it is not difficult to see that $E_{\bullet }$ is freely generated (as a simplicial monoid) by the vertex $e$ and the edge $h: e^2 \rightarrow e$. In particular, the homomorphism of simplicial monoids $\varphi : E_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{Dy} )$ is uniquely determined by the requirement that it carries $e$ to the unit interval $[0,1] \in \mathrm{Dy}$ and $h$ to the dyadic homeomorphism $x \mapsto x/2$.