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Let $\mathcal{C}$ be a category with finite limits and a strictly initial object $\mathbf{0}$. The final object is denoted by $\mathbf{1}$. I propose the following definition of a field object internal to $\mathcal{C}$:

Definition. A field object internal to $\mathcal{C}$ is a commutative ring object $R=(X,+,0,*,1)$ internal to $\mathcal{C}$ with the following property: Let $X^*$ be the equalizer of the maps $X \times X \rightrightarrows X$, defined by $(x,y) \mapsto x \cdot y$ resp. $(x,y) \mapsto 1$ on points. We may write $X^* = \{(x,y) \in X \times X : x \cdot y = 1\}$. Let $p : X^* \to X$ be the projection $(x,y) \mapsto x$. Then, we require that a morphism $T \to X$ factors through $p$ if and only if $T \to X$ and $\mathbf{1} \xrightarrow{0} X$ are disjoint, i.e. $$\begin{array}{cc} \mathbf{0} & \to & \mathbf{1} \\ \downarrow && ~~\downarrow {\scriptsize 0} \\ T & \to & X \end{array}$$ is a pullback square. Notice that this implies, in particular, that $1$ and $0$ are disjoint.

Examples. Field objects internal to $\mathsf{Set}$ coincide with fields in the usual sense, right? Also, field objects internal to $\mathsf{Top}$ should coincide with topological fields in the usual sense. This is a bit subtle, because notice that, for any commutative ring object, we have a well-defined inversion map $X^* \to X^*$, $(x,y) \mapsto (y,x)$. But in the definition of a topological field, one usually assumes continuity of the inversion map on $X \setminus \{0\}$, which is not automatic. The solution is that for topological rings whose underlying ring is a field, the map $X^* \to X \setminus \{0\}$, $(x,y) \mapsto x$ has no reason to be a homeomorphism, unless the inversion map on $X \setminus \{0\}$ is continuous. This is somewhat built into the definition of a field object, namely applied to the inclusion map $X \setminus \{0\} \to X$.

Questions. So this definition of a field object seems reasonable to me. But I have never seen it before. In fact, on mathoverflow there was some discussion that such a definition is either impossible or useless. Nevertheless, do you also think that my definition is correct? Is it natural? What other definitions can you think of? Is there some literature about this?

Edit. In a comment Laurent Moret-Bailly points out that $\mathbb{A}^1_S$ carries a field structure internal to the category of $S$-schemes in the above sense. This is because a global section $s$ of an $S$-scheme $T$ is invertible if and only if the fiber product of $s : T \to \mathbb{A}^1_S$ with the zero section $0 : S \to \mathbb{A}^1_S$ is empty.

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    $\begingroup$ Seems like the definition of a residue field object ncatlab.org/nlab/show/field $\endgroup$ – David Roberts Mar 3 '15 at 10:01
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    $\begingroup$ Several possible versions (most of them needing substantially more than finite limits to formulate) are studied in Johnstone's "Rings, Fields and Spectra" (J. Algebra 49, 1977, 238-260). The version you are interested in must be more or less what Johnstone calls geometric field. $\endgroup$ – მამუკა ჯიბლაძე Mar 3 '15 at 10:02
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    $\begingroup$ It appears to me your definition is a translation of "$x$ is invertible in a field if and only if $x \ne 0$", where you have interpreted $x \ne 0$ using the notion of a complementary subobject. This is a natural definition, I suppose. $\endgroup$ – Zhen Lin Mar 3 '15 at 11:03
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    $\begingroup$ @მამუკაჯიბლაძე No, I don't think Martin's definition is especially similar to the notion of geometric field (which nLab calls "discrete field"). It's much closer to the definition via axiom F2. $\endgroup$ – Zhen Lin Mar 3 '15 at 11:35
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    $\begingroup$ Never thought about this before, but it does sound right. So, for instance, for any scheme $S$, $\mathbb{A}^1_S$ is a field object in the category of $S$-schemes. $\endgroup$ – Laurent Moret-Bailly Mar 3 '15 at 16:28
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It's probably worth pointing out that there are various definitions that are equivalent in $\text{Set}$ but not in general. Let me call your definition invertible-elements. Here is another:

Free-modules: A field $k$ is a commutative ring, not isomorphic to the terminal commutative ring, such that every $k$-module is free (that is, we require that the forgetful functor from $k$-modules down to the underlying category has a left adjoint, and free objects are objects in the essential image of this left adjoint).

This definition doesn't agree with yours in topoi. For example, in the topos of $G$-sets, any field $k$ with trivial $G$-action is an invertible-elements field, but almost never a free-modules field: $k$-modules in $G$-sets are externally $k[G]$-modules, and in general not every $k[G]$-module is the free $k[G]$-module on a $G$-set.

This suggests that free-modules is a bad definition because it doesn't do the right thing intuitionistically (probably a standard observation to the topos theory cognoscenti), but in any case it means that what definition you pick depends on what you want to do with it.

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  • $\begingroup$ Wouldn't we rather every module be projective instead of free? $\endgroup$ – David Roberts Mar 4 '15 at 3:54
  • $\begingroup$ @David: that gives semisimple rings (so in the commutative case, finite direct products of fields). $\endgroup$ – Qiaochu Yuan Mar 4 '15 at 4:09
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Martin's claim seems to be that he can define a field in a category with products and an initial object, but I am skeptical of this. I am uneasy with the language of the definition, in particular with the occurrences of $X\setminus\lbrace 0\rbrace$. This seems to be a mixture of categorical diagrams and a naive mathematical language that possibly assumes excluded middle. I think it would be a good exercise first to try to define an integral domain in this kind of language.

Yves Diers did a lot of work in the 1980s on disjunctive theories in his categorical study of commutative algebra. A useful survey paper that will give you a good introdction to this is A syntactic approach to Diers’ localizable categories by Peter Johnstone in M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, volume 753 of Lecture Notes in Mathematics pp 466–478. Springer Verlag, 1979.

Following the ideas in these papers, one can define a field in an extensive category with products, that is, one can express axioms such as $$ 0=1 \ \vdash\ \bot, \qquad x y = 0 \ \vdash\ x=0 \lor y=0, \qquad \ \vdash\ x=0 \lor \exists y. x y = 1. $$

Any category that one would be likely to regard as a "category of spaces" (in a very general sense, including locales and affine varieties) is extensive. So, if Martin wants to define fields more generally than this, I would first ask why? Does he have in mind some candidate for a "field" in a particular concrete category with products and an initial object but not coproducts?

The foregoing comments assume that we working with fields where equality makes sense, such as algebraic number fields. In the case of $\mathbb R$, by contrast, inequality is the more natural relation. I could say a lot more about this from the point of view of different approaches to constructive analysis, but I haven't studied or thought about is in the context of categorical logic.

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    $\begingroup$ $X \setminus \{ 0 \}$ never appears in the definition; Martin is only using it to discuss an example. $\endgroup$ – Qiaochu Yuan Mar 3 '15 at 19:21

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