directed object (Rev #4)

Let $\Delta$ denote the simplex category. This is the category having finite ordinals as objects and as morphisms monotone maps thereof.

We define the *category of simplicial sets*by $s Set:=Psh(\Delta)$.

Let $\Delta_0$ be the terminal category (the category with one object $*$ and one morphism $id_*$. Then $Psh(\Delta_0)=Set_{disc}$ is the discrete category of sets; this is the class of sets and the class of morphisms consists only of the identities.

Let $Q:=\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}$ denote the category with two objects and morphism set $\{s,t,id_0,id_1\}$. $Q$ is called the *walking quiver*.

A functor $q:\to Set$ is called a *quiver?*. This is just a directed graph perhaps with multiple edges and loops.

We denote the *category of quivers* with natural transformations thereof as morphisms by $Quiver:=Psh(Q)$.

Are there for the objects $X$ in $sSet$, $Quiver$ or $Set_{disc}$ directed past space objects $dX$?

The interval object in any of these categories is $\Delta[1]$. Let $d_{\Delta[1]}\subset [\Delta[1], \Delta[1]]=sSet([1],[1])$