nLab
Grpd

Definition

The (2,1)-category $\mathrm{Grpd}$ is the 2-category whose

• objects are small groupoids;

• morphisms are functors;

• 2-morphisms are natural transformations, which are necessarily natural isomorphisms.

This is the full sub-2-category of Cat on those categories that are groupoids.

Properties

One may regard $\mathrm{Gprd}$ also just as a 1-category by ignoring the natural isomorphisms betweemn functors. This 1-category may be equipped with the natural model structure on groupoids to provide a 1-categorical presentation of the full $(2,1)$-category.

category: category