Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor $$F_U:\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$ I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-colimit of this and obtain an equivalence of categories $$\operatorname{2-colim}_U\textsf{QCoh}(U) \xrightarrow{\sim} \textsf{Vect}(K).$$ If this is true, I wonder moreover if the natural functor $$\textsf{QCoh}(U) \to \textsf{Vect}(K)$$ is the one which sends a quasi-coherent module over $U$ to its stalk on the generic point.

Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor $$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$ I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-colimit of this and obtain an equivalence of categories $$\operatorname{2-colim}_U\textsf{QCoh}(U) \xrightarrow{\sim} \textsf{Vect}(K).$$ If this is true, I wonder moreover if the natural functor $$\textsf{QCoh}(X) \to \operatorname{2-colim}_U\textsf{QCoh}(U)\cong \textsf{Vect}(K)$$ is the one which sends a quasi-coherent module over $X$ to its stalk on the generic point.

Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor $$F_U:\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$ I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-colimit of this and obtain a functor $$\operatorname{2-colim}_U F_U:\textsf{QCoh}(X) \to \textsf{Vect}(K),$$ which should send a quasi-coherent module over $X$ to its stalk on the generic point.

Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor $$F_U:\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$ I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-colimit of this and obtain an equivalence of categories $$\operatorname{2-colim}_U\textsf{QCoh}(U) \xrightarrow{\sim} \textsf{Vect}(K).$$ If this is true, I wonder moreover if the natural functor $$\textsf{QCoh}(U) \to \textsf{Vect}(K)$$ is the one which sends a quasi-coherent module over $U$ to its stalk on the generic point.