This might be a vague question, but I am troubled by the fact that fields do not admit a nifty categorical definition. An obvious attempt such a definition would be to say that fields are commutative groups in $\mathcal{Ab}$, using that rings are monoids. This fails because the tensor product isn't the categorical product and thus doesn't have diagonals.


Can this be fixed using a more general definition of a group object [[1](https://mathoverflow.net/questions/66474/the-definition-of-a-group-object-is-wrong)]
or can the definition of a field be relaxed in a sensible way?


Is this issue with definability related to why the category of fields is badly behaved? (eg. no limits / colimits, lack of a "free field") [[2](http://ncatlab.org/nlab/show/field)]



Fields aren't group objects in Ab, so what are they?
Fields are the [simple objects](http://ncatlab.org/nlab/show/simple+object) in $\text{CRing}$. 


**Edit:** Some philosophical remarks. Elements having inverses is a property and not a structure, so in some sense it's not obviously a good idea to treat the inverse as extra structure. Talking about group objects instead of just monoid objects can really only be done in cartesian monoidal categories; in general you instead want to talk about monoid objects with some extra property. For example, Poisson-Lie groups are not group objects in the category of Poisson manifolds (which is not cartesian monoidal). Similarly, Hopf algebras are not group objects in the category of coalgebras (which is again not cartesian monoidal). 


For commutative rings "$x$ is invertible" should be thought of as "the ideal generated by $x$ is the unit ideal," and of course this holds for all nonzero $x$ if and only if there are no nontrivial quotients. This suggests that if we want to generalize the definition of a field to other categories similar to $\text{CRing}$ then we should try to generalize this condition. 


*Example.* For graded rings the natural generalization is "the *homogeneous* ideal generated by $x$ is the unit ideal." This is true precisely for graded rings such that every nonzero *homogeneous* element is invertible, such as the ring of Laurent polynomials over a field.


