9.3.2.1 Connected Components of Categories

Let $\mathcal{C}$ be a category.

Definition 9.3.2.1.1 ▶ Connected Components of Categories

A connected component of $\mathcal{C}$ is a full subcategory $\mathcal{I}$ of $\mathcal{C}$ satisfying the following conditions:1

  1. Non-Emptiness. We have $\text{Obj}\webleft (\mathcal{I}\webright )\neq \text{Ø}$.
  2. 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 $B$.

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