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