A connected component of $\mathcal{C}$ is a full subcategory $\mathcal{I}$ of $\mathcal{C}$ satisfying the following conditions:1
- Non-Emptiness. We have $\text{Obj}\webleft (\mathcal{I}\webright )\neq \text{Ø}$.
- Connectedness. There exists a zigzag of arrows between any two objects of $\mathcal{I}$.
1In other words, a connected component of $\mathcal{C}$ is an element of the set $\text{Obj}\webleft (\mathcal{C}\webright )/\mathord {\sim }$ with $\mathord {\sim }$ the equivalence relation generated by the relation $\mathord {\sim }'$ obtained by declaring $A\sim ' B$ iff there exists a morphism of $\mathcal{C}$ from $A$ to $B$.
