Homology and Cohomology Viewed as Functors

One of the reasons why homology groups play such a central role in many contexts is that the whole construction can be viewed as a certain functor. 

If we consider cohomology groups instead, we will get a classical example of a contravariant functor. 

Proposition 4.32. For any commutative ring TZ and any d 0, we have a contravariant functor ASC — Ab mapping an abstract simplicial complex A to H A TZ). 

We just need to see that a simplicial map f Ai — A2 induces an 7 .-module homomorphism / H A2 TZ) — H A ]TZ) which satisfies the properties of the contravariant functor. This can be done by dualizing the proof of Proposition 3.33.  

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