Is an abstract simplicial complex a quiver? Let $\Delta$ be an abstract simplicial complex. Then for $B\in \Delta$ and $A\subseteq B$ we have that $A\in\Delta$. If we define $V$ to be the set of faces of $\Delta$, construct a directed edge from $B$ to $A$ if $A$ is a face of $B$ (i.e. $A\subseteq B$) and define $E$ to be the set of directed edges, then will $\Gamma=(V,E)$ be a quiver?

Yes, and it's the poset of faces ordered by inclusion.
