+-- {: .num_defn} ###### Definition (some toposes of arrows) 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 $Quiver:=Psh(Q)$. =-- The [[interval object]] in any of these categories is $\Delta[1]$ +-- {: .num_defn} ###### Definition (object with directed pathspace) Let $C$ be a category with an [[interval object]] $I$, and suppose that every object $X$ of $C$ is $I$-[[undirected object|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: 1. (Constant paths) Every map $I \to \pt \to X$ is directed; 1. (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 $d_X$; 1. (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 $d_I{C}$. =--