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.

residue fieldobject ncatlab.org/nlab/show/field $\endgroup$4more comments