Can groups be recovered as "monoids" in a bicategory?

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?

• Welcome back, Joey! Mar 22, 2019 at 19:45

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.
• BTW, such an $(X,M)$ is usually called a monad in $\cal C$ (ncatlab.org/nlab/show/monad). Mar 22, 2019 at 19:06
• @JoeyHirsh The last paragraph answers your question. Take the bicategory with a single object $*$ and $\mathrm{Hom}(*,*)$ the category of groups. The composition is the coproduct of groups. Then the category of monads in this bicategory is equivalent to the category of groups. Mar 23, 2019 at 6:51