Authors: Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: On a cohomology theory based on hyperfinite sums of microsimplexes
Journal: Pacific Journal of Mathematics
Volume: 128
Issue: 1
First page: 201
Last page: 208
Issue Date: 1-Jan-1987
Rank: M23
ISSN: 0030-8730
DOI: 10.2140/pjm.1987.128.201
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.
Publisher: MPS

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