Overview

Definition

Recall the definitions of cycles and boundaries:

Definition of cycles

Let be a chain complex. Then there is a graded abelian group with

These are called cycles.
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Definition of boundaries

Let be a chain complex. There is a graded abelian group with

These are called boundaries.
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There is a graded abelian group with

is called the n-th homology group of A.

Or written more succintly, the homology of a complex is the graded abeilian group (or R-module) with

Let be the category of chain complexes over with morphisms as chain maps. Homology defines a functor

to the category of graded -modules.

Chain maps descend to homotopy such that for a map we have,

In order to prove that taking homology is a functor, we first will show that for complexes and chain maps and ,

For the left-hand side we can take as the function that maps from to by . Hence,

Next, doing the composition on the right, we have,

Therefore, the proposed functor respects composition. Lastly, consider , where,

for all . Then using above result from lecture again, we have,

which is the identity map on the graded -modules that are mapped to under the functor. Therefore, is a functor.