Basically bounded functors and flat sheaves

Abstract

A functor is here called basically bounded if, roughly speaking, it is determined by its values on objects of some bounded cardinality. For functors on R-algebras, it is shown that common constructions involving basically bounded functors can again be computed on algebras of bounded size, and hence are uniquely defined irrespective of any special set-theoretic assumptions. Even operations which seem to require arbitrarily large algebras—computing Čech cohomology and sheafifications in the flat topology, forming Ext groups and sheaves—turn out to be basically bounded. The proofs use homological algebra and a notion of approximation by small coverings. © 1975 Pacific Journal of Mathematics Manufactured and first issued in Japan.