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 \emptyset $.
- Connectedness. There exists a zigzag of arrows between any two objects of $\mathcal{I}$.