I have a group with a category structure (i.e. category whose objects form a group), such that the left multiplication with any fixed element is a category automorphism. The same is true for right multiplication. Is there a well-known name for that kind of groups? Any references?
If the category can be interpreted as an ordered set, the group is called ordered group. So what is the name of the generalisation to categories?
Note: For those who consider groups as one-object categories: In this setting I'd like to know how a 2-category is called that is a group.
Edit: Clarify group with category structure.
The correct definition requires some care around what it means for such a thing to be associative. It turns out that associativity requires extra data: you are looking for a monoidal category, which involves extra maps called associators, and you also want the monoidal structure to be grouplike, which means that every object is invertible. So I would call such a thing a grouplike monoidal category. It's somewhat awkward terminology but all of the other options I know are similarly awkward.
Every monoidal category gives rise to a grouplike monoidal category given by taking the subcategory of invertible objects. A simple example is the category of $R$-modules for $R$ a commutative ring; taking invertible objects gives the grouplike monoidal category of invertible $R$-modules, or line bundles over $\text{Spec } R$.