The following is probably a bad question, but hopefully, it might have a very good answer.
In category theory there is a quite famous analogy between topoi and commutative rings, I was never convinced by this analogy, but the best way to see how far an analogy can be pushed is to challenge it. Clicking on the link that I provided above you can have an extensive presentation of the analogy, the general motto can be grasped by the following table.
Remark 6.1.1.3. $\space$ Let $\mathcal{X}$ be an $\infty$-category. The assumption that colimits in $\mathcal{X}$ are universal can be viewed as a kind of distributive law. We have the following table of vague analogies:
$$\begin{array}{ccc} && \text{Higher Category Theory} && \quad && \text{Algebra} && \\ \hline \\ & & \infty\text{-Category} & & & & \text{Set} \\ \\ & & \text{Presentable } \infty\text{-category} & & & & \text{Abelian group} \\ \\ & & \text{Colimits} & & & & \text{Sums} \\ \\ & & \text{Limits} & & & & \text{Products} \\ \\ & & \varinjlim(X_\alpha) \times_S T \simeq \varinjlim(X_\alpha \times_S T) & & & & (x + y)z = xz + yz \\ \\ & & \infty\text{-Topos} & & & & \text{Commutative ring} \end{array}$$ Definition 6.1.1.2 has a reformulation in the language of classifying functors ($\S$3.3.2):
That corresponds to Rem 6.1.1.3 in my version of HTT by Lurie.
Q. According to this analogy, what should be a field?
Maybe I should say why this might be a stupid question or even a stupid challenge for the analogy. In fact it might be the case that:
- The notion of field is interesting only in low dimension.
- The correct generalization of the notion of field looks very different in categories and trivializes for sets because of their intrinsic rigidity.
