TITLE: Fields aren't group objects in Ab, so what are they?
QUESTION [12 upvotes]: 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]
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]

REPLY [8 votes]: Perhaps it is better to think of fields as particularly nice local rings. Local rings can be defined internal to a topos, and indeed this is (EDIT: related to) why stalks of sheaves of rings field-valued functions (EDIT: on schemes) turn out to be local rings. As far as a category-theoretic view on fields, there are several ways to look at them, as discussed at the nLab entry field in the section 'constructive notions'; note that here one can essentially ignore the constructivism, and think of these as different ways to specify a field object in a category (with appropriate structure/properties). The categories of various types of internal fields can given by the categories of models for various limit-colimit sketches, with the different notions given by different sketches. Group and ring objects are given by models of Lawvere theories, which are sketches using only cones over discrete diagrams - too simple to give the full definition of a field.