nLab
Grp

Definition

$\mathrm{Grp}$ is the category with groups as objects and group homomorphisms as morphisms.

More abstractly, we can think of $\mathrm{Grp}$ as the full subcategory of $\mathrm{Cat}$ with groups as objects.

Remarks

Since groups may be identified with one-object groupoids, it is sometimes useful to regard $\mathrm{Grp}$ as a $2$-category, namely as the full sub-$2$-category of Grpd on one-object groupoids. In this case the $2$-morphisms between homomorphisms of groups come from “intertwiners”: inner automorphisms of the target group.

On the other hand, if we regard $\mathrm{Grp}$ as a full sub-$2$-category of ${\mathrm{Grpd}}_{*}$, the $2$-category of pointed groups, then this is locally discrete and equivalent to the ordinary $1$-category $\mathrm{Grp}$. This is because the only pointed intertwiner between two homomorphisms is the identity.

Precisely analogous statements hold for the category Alg of algebras.

category: category