This is a follow up question to this question.
Is every scheme over a field $K$ the colimit (over some arbitrary complicated diagram) of affine schemes $\operatorname{Spec}(R_\alpha)$ where each $R_\alpha$ is a $K$-algebra and a local ring?
Intuitively, my question is ''scheme $=$ local rings $+$ glueing information?''.
EDIT: Sorry, I misread w/c category you were asking had the colimit. The schemes are at least sort of colimits of the spectra, because each $\mathscr{O}_X(X)$ is a certain limit (the inverse limit) of $O_X$'s rings at its basis sets.
You don't glue the open sets as colimits of the stalks, you glue them as inverse limits over the sets within it that are in a basis of open sets. See the gluing axiom for sheaves, in particular using B-sheaves. Since the restriction map sends $\mathscr{O}(U)$ to $\mathscr{O}(x)$ for every neighborhood $U$ of $x$, the coproduct over all the stalks or values over the basis is a good start, but the question is what's the quotient from the coproduct to the inverse limit. See Geometry of Schemes for a full explanation.