Is there a bicategory $V$ and a definition of *monoid in a bicategory* so that $\text{Monoids}(V)$ is the category of groups and homomorphisms?


EDIT: For example, is there a bicategory $V$ so that Monad(V) is the category of groups and group homomorphisms?



Can groups be recovered as "monoids" in a bicategory?
There is a definition of monoids in a monoidal category and every monoidal category is a bicategory. You can try to extend this definition to a general bicategory. For example, we can say that a monoid in a bicategory $\mathcal{C}$ is a monoid in the monoidal category $\mathrm{Hom}\_\mathcal{C}(X,X)$ for every object $X$ of $\mathcal{C}$ or that a monoid is a pair $(X,M)$, where $X$ is an object of $\mathcal{C}$ and $M$ is a monoid in $\mathrm{Hom}\_\mathcal{C}(X,X)$. I'm not sure what are you looking for, so I can't say if these definition are what you need.


Anyway, any category $\mathcal{C}$ with finite coproducts is a monoidal category and the category of monoids in such a category is $\mathcal{C}$ itself. So, you can take $V$ to be the bicategory corresponding to the monoidal category of groups.


