the definition of a residue field in a scheme as a functor We define a functorial scheme as in "Two functorial definitions of schemes". We would like to define a residue field of a scheme. The first idea I came up with was to use a notion of a structure sheaf. As is well known, the one of the way to define a residue field is to use the stalk of a structure sheaf. A structure sheaf can be defined as this MSE answer in a functorial scheme. However, to get the stalk require a set theoretical point, and so we don't know set theoretical points of a functor. This is why we can't get the stalk of a structure sheaf. Can we define the definition of a residue field in a scheme as a functor?

If $X$ is a scheme and $p\in X(k)$ is a $k$-point for some field $k$, you can define the residue field at $p$ as the smallest subfield $k_0\subseteq k$ such that $p$ lifts to a point of $X(k_0)$ (with respect to the inclusion map $k_0\to k$).
