Hyperfinite constructions of nonstandard analysis were used in homology and cohomology theory by M. C. McCord, Rade Živaljević and Jerome Keisler (wikipedia, homepage).
See
Rade T. Živaljević, On a cohomology theory based on hyperfinite sums of microsimplexes, Pacific J. Math. Volume 128, Number 1 (1987), 201-208; (project euclid)
M. C. McCord, Non-standard analysis and homology, Fund. Math., 74 (1972), 21-28; (pdf)