Kerodon

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Construction 2.3.6.4. Let $(Q, \leq )$ be a partially ordered set, which we regard as a category having a unique morphism $e_{y,x}$ for every pair of elements $x,y \in Q$ with $x \leq y$. We define a strictly unitary lax functor $T_{Q}: Q \rightarrow \operatorname{Path}_{(2)}[Q]$ as follows:

  • On objects, the lax functor $T_{Q}$ is given by $T_{Q}(x) = x$.

  • On $1$-morphisms, the lax functor $T_{Q}$ is given by $T_{Q}( e_{y,x} ) = \{ y,x \} \in \underline{\operatorname{Hom}}_{ \operatorname{Path}_{(2)}[Q]}(x,y)$ whenever $x \leq y$ in $Q$.

  • For every triple of elements $x,y,z \in Q$ satisfying $x \leq y \leq z$, the composition constraint $\mu _{z,y,x}: T_{Q}( e_{z,y} ) \circ T_{Q}( e_{y,z} ) \Rightarrow T_{Q}( e_{z,x} )$ is the $2$-morphism of $\operatorname{Path}_{(2)}[Q]$ corresponding to the inclusion of linearly ordered sets

    \[ T_ Q( e_{z,x} ) = \{ z, x \} \subseteq \{ z, y, x \} = \{ z, y \} \cup \{ y, x \} = T_{Q}( e_{z,y} ) \circ T_{Q}( e_{y,x} ). \]