Construction 8.3.3.1.3 (00ZL): Construction of the Core of a Category

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Table of Contents
Part 3: Category Theory
Chapter 8: Categories
Section 8.3: Groupoids
Subsection 8.3.3: The Core of a Category
Construction 8.3.3.1.3: Construction of the Core of a Category (cite)

Construction 8.3.3.1.3 ▶ Construction of the Core of a Category

The core of $\mathcal{C}$ is the wide subcategory of $\mathcal{C}$ spanned by the isomorphisms of $\mathcal{C}$, i.e. the category $\mathsf{Core}\webleft (\mathcal{C}\webright )$ where ^[1]

Objects. We have
\[ \text{Obj}\webleft (\mathsf{Core}\webleft (\mathcal{C}\webright )\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Obj}\webleft (\mathcal{C}\webright ). \]
Morphisms. The morphisms of $\mathsf{Core}\webleft (\mathcal{C}\webright )$ are the isomorphisms of $\mathcal{C}$.

Proof of Construction 8.3.3.1.3.

This follows from the fact that functors preserve isomorphisms (Item 1 of Proposition 8.4.1.1.6).

Footnotes

[1] Slogan: The groupoid $\mathsf{Core}\webleft (\mathcal{C}\webright )$ is the maximal subgroupoid of $\mathcal{C}$.

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