Proposition 11.1.3.1.3 (00XL): Properties of Skeletons of Categories

Table of Contents
Part 4: Categories
Chapter 11: Categories
Section 11.1: Categories
Subsection 11.1.3: Skeletons of Categories
Proposition 11.1.3.1.3: Properties of Skeletons of Categories (cite)

11.1.3 Skeletons of Categories

Definition 11.1.3.1.1 ▶ Skeletons of Categories

A¹ skeleton of a category $\mathcal{C}$ is a full subcategory $\mathsf{Sk}\webleft (\mathcal{C}\webright )$ with one object from each isomorphism class of objects of $\mathcal{C}$.

¹Due to Item 3 of Proposition 11.1.3.1.3, which states that any two skeletons of a category are equivalent, we often refer to any such full subcategory $\mathsf{Sk}\webleft (\mathcal{C}\webright )$ of $\mathcal{C}$ as the skeleton of $\mathcal{C}$.

A category $\mathcal{C}$ is skeletal if $\mathcal{C}\cong \mathsf{Sk}\webleft (\mathcal{C}\webright )$.¹

¹That is, $\mathcal{C}$ is skeletal if isomorphic objects of $\mathcal{C}$ are equal.

Proposition 11.1.3.1.3 ▶ Properties of Skeletons of Categories

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

1.
Existence. Assuming the axiom of choice, $\mathsf{Sk}\webleft (\mathcal{C}\webright )$ always exists.
2.
Pseudofunctoriality. The assignment $\mathcal{C}\mapsto \mathsf{Sk}\webleft (\mathcal{C}\webright )$ defines a pseudofunctor

\[ \mathsf{Sk}\colon \mathsf{Cats}_{\mathsf{2}}\to \mathsf{Cats}_{\mathsf{2}}. \]
3.
Uniqueness Up to Equivalence. Any two skeletons of $\mathcal{C}$ are equivalent.
4.
Inclusions of Skeletons Are Equivalences. The inclusion

\[ \iota _{\mathcal{C}}\colon \mathsf{Sk}\webleft (\mathcal{C}\webright )\hookrightarrow \mathcal{C} \]

of a skeleton of $\mathcal{C}$ into $\mathcal{C}$ is an equivalence of categories.