Subsection 9.4.2 (01U2): Groupoids

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A groupoid is a category in which every morphism is an isomorphism.

The isomorphism of categories of Example 9.2.3.1.1 restricts to an isomorphism

where $\mathsf{Grpd}$ is the full subcategory of $\mathsf{Cats}$ spanned by the groupoids.

In other words, we have an identification

\[ \webleft\{ \text{Groups}\webright\} \cong \webleft\{ \text{One-object groupoids}\webright\} . \]

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