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Aug 2, 2022 at 13:33 comment added Qiaochu Yuan Seems to me like a benefit of this point of view is that the construction of the absolute Galois group requires some use of choice but constructing and discussing functors on étale algebras requires no choice (that I know of).
Aug 2, 2022 at 9:13 comment added Neil Strickland For example, given an elliptic curve $E$ you can consider the Galois representation $E(\overline{K})$ or the functor $A\mapsto E(A)$.
Aug 2, 2022 at 8:54 comment added Kim This sounds like an interesting interpretation. Could you elaborate with an example of an $X$ where we can see an explicit $FX$ and $Y$?
Aug 2, 2022 at 8:46 history answered Neil Strickland CC BY-SA 4.0