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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]

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    $\begingroup$ It can still be done. You just need more diagrams to ensure you have a 1, that $0\neq 1$, distributivity, etc. If I have time later I'll upload an answer. In short, fields will be field-objects in the category of sets. $\endgroup$
    – David White
    Commented Nov 15, 2013 at 21:05
  • $\begingroup$ Related: mathoverflow.net/questions/3003 $\endgroup$
    – Martin Brandenburg
    Commented Nov 17, 2013 at 8:02

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Fields are the simple objects 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.

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    $\begingroup$ nLab defines simple objects for categories with zero objects, and the considers the category of unital rings with $0\neq 1$, which has no zero object. $\endgroup$
    – Fernando Muro
    Commented Nov 15, 2013 at 21:44
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    $\begingroup$ Hmm. Well, we get a zero object if we move to non-unital rings so this doesn't seem like a big deal. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 15, 2013 at 21:52
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    $\begingroup$ @Kris: the zero ring is not a zero object in the category of unital rings. It is only the terminal object, while the initial object is $\mathbb{Z}$. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 15, 2013 at 23:39
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    $\begingroup$ As for the last paragraph: Actually there is a well-known notion of "graded field" (defined for example by the condition that every graded module is graded-free), and every such is isomorphic to $K[x,x^{-1}]$ for some field $K$. $\endgroup$
    – Martin Brandenburg
    Commented Nov 17, 2013 at 8:03
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    $\begingroup$ "if we want to generalize the definition of a field to other categories similar to CRing" ... we could look for example at Yves Diers book Categories of commutative algebras, where field objects in arbitrary so-called Zariski categories are defined. Actually just by the condition that they are simple. But the nlab article on fields also explains how to internalize the notion of a field using pullbacks. $\endgroup$
    – Martin Brandenburg
    Commented Nov 17, 2013 at 8:06
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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.

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    $\begingroup$ "this is why stalks of sheaves of rings turn out to be local rings" Well this is not true. This the difference between locally ringed spaces and ringed spaces (and as usual you may replace "space" by "topos" here). $\endgroup$
    – Martin Brandenburg
    Commented Nov 17, 2013 at 8:00
  • $\begingroup$ I probably should have said 'on schemes'. Also, I was thinking of the specific case of field-valued functions - so I shall edit it shortly. $\endgroup$
    – David Roberts ♦
    Commented Nov 17, 2013 at 20:39

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