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.
