Categorical characterisation of fields and groups

Question

Given a monoidal category $\mathcal{V}$, is there a categorical characterisation of an object in $\mathcal{V}$ such that it is a group if $\mathcal{V} = Set$ and a field if $\mathcal{V} = Ab$? Or does anybody know of useful references? This would be much appreciated!

Ofcourse I do not expect it by simple means of extra data as groups cannot be descriped operadically, but rather by universal properties or something similar.

EDIT: As noted in the comments: I am looking for something different than group objects, as they do not encompass fields. So I am looking for some categorical framework in which both can fit.

Answer 1

As far as I know there is no such characterization. Groups are, of course, the group objects in $\text{Set}$; to my mind the cleanest way to characterize fields is that they are the simple objects in the category of commutative rings (the ones with no nontrivial quotient objects, where by "quotient object" I mean "effective epimorphism.")

Groups and fields are just not actually that analogous. Fields are required to be commutative and inverses are only defined for nonzero elements, so they aren't actually "algebraic." And there's no way to talk about group objects in a monoidal category that isn't cartesian; the closest you get is talking about Hopf algebras, which are very different from fields.

Answer 2

For groups, it's quite doable: a group object in a cartesian monoidal category is a particular kind of monoid. Now every object in a cartesian monoidal category is canonically a comonoid, where the comultiplication is the diagonal $\Delta:G\to G\times G$ and the counit is the unique map $!:G\to 1$. In fact if $G$ is a monoid, then this comonoid structure makes it a bimonoid, because $\Delta$ and $!$ are always monoid homomorphisms.

This suggests that we want to think about a kind of bimonoid, in a general symmetric monoidal category (the symmetry is necessary to make the tensor product of monoids a monoid.) So let $G$ in $\mathcal V$ be a bimonoid, equipped with a comultiplication $\Delta:G\to G\otimes G$ and a counit $\epsilon:G\to I$ as well as a multiplication $\mu$ and unit $\eta$. Then we can write down the inverse axiom as for a group: introduce an endomorphism $s:G\to G$ (called in this context an "antipode"), such that $\mu\circ(1\otimes s)\circ \Delta=\eta\circ \epsilon$. This is the definition of a Hopf monoid, and uniqueness of comonoid structures in Cartesian monoidal categories implies that a Hopf monoid in Sets is precisely a group.

I'm not so sure what to do about fields, since these aren't even algebraically defined--note that we need the notion of the complement of zero to get a field, so they naturally fit in something closer to a topos rather than in a monoidal category.