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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 setsby $s Set:=Psh(\Delta)$.
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 $Quiver:=Psh(Q)$.
The Are there for the objectsinterval object$X$ in any of these categories is$\mathrm{\Delta sSet}[1]$ \Delta[1] sSet, $Quiver$ or $Set_{disc}$ directed past space objects $dX$?
Let $C$ be a category with an interval object $I$, and suppose that every object $X$ of $C$ is $I$-undirected?.
To be explicit, fix a subset $d_I \subset {}_{pt}hom(I,I)_{pt}$ of the endomorphisms of the given interval object $I$ regarded as a cospan $pt \to I \leftarrow pt$ to be called the directed endomorphisms of the interval object. Let $d_X\subset [I,X]$ be a subset of the hom-set? $[I \to X]$.
Then we call the pair $(X, d_X)$ an object with directed path space $d_X$ if the following conditions (attributed to Marco Grandis) are satisfied:
(Constant paths) Every map $I \to \pt \to X$ is directed;
(Reparametrisation) For $\gamma \in d_X \subset hom(I,X)$ and every $\phi \in d_I \subset hom(I,I)$, also $\gamma \circ \phi$ is in $d_X$;
(Concatenation) Let $a,b:I\to X$ be consecutive wrt. $I$ (i.e. $\pt \to^{\tau} I \to^{a} X$ equals $\pt \to^{\sigma} I \to^{b} X$), let $I^{v2}$ denote the pushout of $\sigma$ and $\tau$, then by the universal property of the pushout there is a map $\phi:I^{v2}\to X$. By definition of the interval object (described there in the section “Intervals for Trimble $\omega$-categories”) there is a unique morphism $\psi:I\to I^{v2}$. Then the composition of $a$ and $b$ is defined by $a\bullet b:=\phi\circ \psi$. Then $d_X$ shall be closed under composition of consecutive paths.
We define a morphism of objects with directed path space to be a morphism of their underlying objects that preserves directed paths. Objects with directed path space and morphisms thereof define a category denoted by $d_I{C}$.
$C$ is a subcategory of $d_I{C}$.
The interval object in any of these categories is $\Delta[1]$. Let $d_{\Delta[1]}\subset [\Delta[1], \Delta[1]]=sSet([1],[1])$