Spahn directed object (Rev #3, changes)

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 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 $Quiver:=Psh(Q)$.

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?.

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;

2. (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 $d_X$;

3. (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 $d_I{C}$.