nLab
hom-groupoid

Idea

A $V$-enriched category $C$, for $V=$ Grpd (the category of groupoids), has for every ordered pair $x,y$ of objects a groupoid $C(x,y)$ of morphisms between $x$ and $y$. This hom-object is hence a hom-groupoid in this case.

For this reason such a category may be thought of as a locally groupoidal 2-category, or (2,1)-category.

Examples

• For $C=$ Grpd the hom-groupoids are the functor categories between two groupoids.

• For $C$ any small category, the (2,1)-presheaf-category $[C^{\mathrm{op}},\mathrm{Grpd}]$ has as hom-groupoid $[C^{\mathrm{op}},\mathrm{Grpd}](A,B)$ the groupoid of pseudonatural transformations and modifications between the pseudo-functors $A,B$.