On a cohomology theory based on hyperfinite sums of microsimplexes

Abstract

In this note we investigate a cohomology theory H#(X, G), defined by M. C. McCord, which is dual to a homology theory based on hyperfinite chains of miscrosimplexes. We prove that if X is a locally contraction, paracompact space then H#(X, G) ≃ Hč#(X, Hom(*Z, G)) where Hč# is the Čech theory. Nonstandard analysis, particularly the Saturation Principle, is used in this proof in essential way to construct a fine resolution of the constant sheaf X × Hom(*Z, Z). This gives a partial answer to a question of McCord. Subsequently, we prove a proposition from which it is deduced that Hom(*Z, Z) = {0} i.e. H#(X, Z) = {0} if X is paracompact and locally contractible. At the end we briefly discuss a related cohomology theory which is obtained by application of the internal (rather than external) Hom(·, G) functor.