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.



Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?
Let $x$ be a point in a scheme $X$. There are two posets, namely the poset of affine opens containing $x$, $A(x)$, and the poset of opens containing $x$, $O(x)$.


The inclusion $A(x)^{op} \to O(x)^{op}$ is (homotopy) cofinal - by Quillen's theorem A, it suffices to show that for any open $U$, $A(x)\_{/U}$ is weakly contractible. But it is a co-directed poset : for any $V,W \to U$, $V\cap W$ is an open and therefore contains some affine open $x\in O\subset V\cap W\subset U$; therefore it is weakly contractible.


It follows that, in the $(2,1)$-category of *presentable* $1$-categories, $colim\_{U\in O(x)^{op}}QCoh(U) \simeq colim\_{U\in A(x)^{op}} QCoh(U)$.


Now note that the latter diagram in fact lives in the category of $E\_0$ *presentable* categories, that is, presentable categories with a chosen basepoint (namely $O\_U$) that are equivalent to something of the form $(Mod\_R,R)$ for some (commutative) ring $R$.


By a $(2,1)$-categorical analogue of theorem 4.8.5.11. in Lurie's *Higher algebra* (which says that $R\mapsto (Mod\_R,R)$, as a functor from rings to $E\_0$ *presentable* categories, is fully faithful and colimit preserving) (it's actually not an analogue, it follows strictly from this theorem by specializing to $Set$-modules in presentable $(\infty,1)$-categories), and using the fact that $A(x)$ is weakly contractible, so colimits over it in $E\_0$-objects are computed underlying, it follows that $colim\_{U\in A(x)^{op}} QCoh(U)\simeq colim\_{U\in A(x)^{op}} Mod\_{O\_U(U)}$ is equivalent to modules over $colim\_{U \in A(x)^{op}} O\_U(U)= colim\_{U\in A(x)^{op}} O\_X(U)= O\_{X,x}$.


Note that here the colimit is taken in ordinary associative rings, but it's a filtered colimit, so it can also be taken in commutative rings.


In conclusion, for any point $x$, $colim\_{U\in O(x)^{op}} QCoh(U)$ is equivalent to modules over $O\_{X,x}$.


If $x$ is a generic point, then $O(x)$ is the category of all opens, and so you get the corresponding statement for $colim\_U QCoh(U)$.


Note that here I've used the following version of "$2-\mathrm{colim}$": homotopy colimits in the $(2,1)$-category (in the sense of "$(\infty,1)$-categories with $1$-truncated mapping spaces") of *presentable* $1$-categories. I'm assuming that this shouldn't matter too much


