I have learned the basic things about cohomology and homology. It seems that homology and cohomology both deal with the same objects, the complexes, but with a different choice of the indexes (for the homology the indexes are decreasing and in cohomology are increasing).
Why, in some geometric applications it is done a radical selection and it is important to distinguish homology from cohomology?
For example if one constructs a Cech homology instead af Cech cohomology, what are the differences?