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**Example text**

In order to be able to compare functors, recall the definitions of the natural transformation of functors and of the equivalence of functors. Let F1 , F2 be functors from a category G1 to a category G2 . e. the following diagram is commutative. F1 (A) F1 (f ) ϕA F2 (A) / F1 (B) ϕB F2 (f ) / F2 (B) 38 1 Elements of homology theory In this case it is said that there is a natural transformation of the functor F1 into the functor F2 . If for any object A in G1 the morphisms ϕA are isomorphisms, then it is said that the functors are equivalent.

The further proof is decomposed into several steps, stated below as a string of exercises. The exercises can also be solved via the method of diagrammatic search. ✷ Exercise 41. Prove that the element h¯ does not depend on the choice of t ∈ p −1 (a). Exercise 42. Prove that the element h¯ does not depend on the choice of the cycle a representing h. Exercise 43. Prove that the sequence of homology groups thus obtained is exact. 36 1 Elements of homology theory It should be noticed that the homomorphisms defined in the proof of Theorem 16 are functorial in the following sense.

Permuting two base elements. The answer is simple: the matrix is subjected to the analogous transformations of columns. If an elementary transformation is applied to the basis of the group Cn−1 , then An undergoes a certain transformation of rows. Reversing these observations yields an easy way of calculating the nth homology group of the complex C. Its essence is to perform simultaneous transformations of the matrices An+1 and An . Step 1. Apply to An elementary transformations of rows and columns bringing it to a diagonal form.