graph

• a system of vertices and arrows
• taken to be directed
• definition: a graph $G$ is
  • a sequence $G := (V, A, src, tgt)$, where:
  • $V$: the set of vertices of $G$
  • $A$: the set of arrows of $G$
  • $src: A \to V$: the source function for $G$
  • $tgt: A \to V$: the target function for $G$

Categories have underlying graphs

• we have a functor: $U: \mathbf{Cat} \to \mathbf{Grph}$
• $(\mathbf{Ob(\mathcal{C})}, \text{Hom}_{\mathcal{C}}, dom, cod)$ is a graph
• a functor $F: \mathcal{C} \to \mathcal{D}$ induces a graph morphism $U(F): U(\mathcal{C}) \to U(\mathcal{D})$