directed object (Rev #2, changes)

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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.

~~ Let~~ We define the~~$X:\Delta^{op}\to Set$~~*category of simplicial sets*~~ ~~ by~~ be~~~~ a~~~~simplicial set?~~$s Set:=Psh(\Delta)$~~. The category of simplicial sets we denote 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 $q:\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}\to Set$ is called a *quiver?*. This is just a directed graph perhaps with multiple edges and loops.

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