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][1], 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.[![enter image description here][2]][2]

That correspond to Rem 6.1.1.3 of 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:

1. The notion of field is interesting only in *low dimension*.
2. The correct generalization of the notion of field looks very different in categories and trivializes for sets because of their intrinsic rigidity.

  [1]: https://www.youtube.com/watch?v=wG5MZqj_JK8
  [2]: https://i.sstatic.net/jhvXv.png