Subsubsection 8.2.2.1 (00Y5): Connected Components of Categories

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Table of Contents
Part 3: Category Theory
Chapter 8: Categories
Section 8.2: The Quadruple Adjunction With Sets
Subsection 8.2.2: Connected Components and Connected Categories
Subsubsection 8.2.2.1: Connected Components of Categories (cite)

8.2.2.1 Connected Components of Categories

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

Definition 8.2.2.1.1 ▶ Connected Components of Categories

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

Non-Emptiness. We have $\text{Obj}\webleft (\mathcal{I}\webright )\neq \emptyset $.
Connectedness. There exists a zigzag of arrows between any two objects of $\mathcal{I}$.

Footnotes

[1] In 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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