A criterion for semiampleness

We suggest a sufficient condition for the existence of a morphism from a diagram of quasipolarized primary algebraic spaces into a polarized pair. Moreover, we describe diagrams in the category of quasipolarized algebraic spaces such that every finite subdiagram of such a diagram has a morphism into a polarized pair and all fine subdiagrams which are closed under inclusions and under skrepas have a polarized colimit. Such diagrams are called sobors, and their arrows are inclusions and skrepas. The main application is a criterion for the semiampleness of a nef invertible sheaf on a complete algebraic space in terms of a sobor.