Grp

**$Grp$** is the category with groups as objects and group homomorphisms as morphisms.

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

Since groups may be identified with one-object groupoids, it is sometimes useful to regard $Grp$ 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 $Grp$ as a full sub-$2$-category of $Grpd_*$, the $2$-category of *pointed* groups, then this is locally discrete and equivalent to the ordinary $1$-category $Grp$. 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

Revised on June 13, 2013 17:27:08
by Urs Schreiber
(82.169.65.155)