Consider a category whose objects are elements of the set $\mathbb{R}^2$. Consider two points $A=(0, 0)$ and $B=(1, 1)$. Define $Hom(A, B)$ to be the set with just one element, the vector $\vec{AB}$. Composition of morphisms would be just addition of vectors and identity would just be the zero vector.
This seems like it is a proper category. Is there name for such a construction in the literature?
It kind of feels like this construction will work with any group, not necessarily a vector space. If I take a group $G$, the category that I'll end up with will be a groupoid, where $\operatorname{Hom}(g,h) = \{g^{-1}h\}$. Every morphism is invertible, justifying me calling the category a groupoid.
I'm not very familiar with category theory, but this seems somewhat like it should be called the discrete groupoid associated to a group $G$: every morphism is invertible and morphisms are uniquely determined by their endpoints.
ETA – @CliveNewstead points out that this is actually a codiscrete groupoid.