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I am curious as to the reasons why one chooses covariant vs. contravariant theories; specifically, I see mention of DeRham Cohomology and Cech Homology, but I rarely see mention of the covariant counterparts DeRham and Cech homology theories.

I think one uses DeRham Cohomology , because it deals with differential n-forms, and n-forms pullback contravariantly, i.e., given a smooth map F: M-->N between manifolds, we get a pullback:

F* : N* -->M* , where N*, M* are the respective dual spaces of N, M. Something similar is the case for the double-, triple- , etc. duals, all of which pullback contravariantly.

Now, how to explain that Cech cohomology is more common than Cech homology? I guess this has to see with properties of sheafs. Now I know relatively little about sheaves. Is this the reason

for using cohomology? If not, what is the reason?

Thanks.

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# What About Cech Homology?

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