Homology functor

A functor on an Abelian category that defines a certain homological structure on it. A system $ H = {( H _ {i} ) } _ {i \in \mathbf Z } $ of covariant additive functors from an Abelian category $ {\mathcal A} $ into an Abelian category $ {\mathcal A} _ {1} $ is called a homology functor if the following axioms are satisfied.

1) For each exact sequence

$$ 0 \rightarrow A ^ \prime \rightarrow A \rightarrow A ^ {\prime\prime} \rightarrow 0 $$

and each $ i $, in $ {\mathcal A} $ a morphism $ \partial _ {i} : H _ {i+ 1} ( A ^ {\prime\prime} ) \rightarrow H _ {i} ( A ^ \prime ) $ is given, which is known as the connecting or boundary morphism.

2) The sequence

$$ \dots \rightarrow H _ { i + 1 } ( A ^ \prime ) \rightarrow H _ {i + 1 } ( A) \rightarrow \ H _ {i + 1 } ( A ^ {\prime\prime} ) \rightarrow ^ { {\partial _ i } } $$

$$ \rightarrow ^ { {\partial _ i} } H _ {i} ( A ^ \prime ) \rightarrow \dots , $$

called the homology sequence, is exact.

Thus, let $ {\mathcal A} = K( \mathop{\rm Ab} ) $ be the category of chain complexes of Abelian groups, and let $ \mathop{\rm Ab} $ be the category of Abelian groups. The functors $ H _ {i} : K( \mathop{\rm Ab} ) \rightarrow \mathop{\rm Ab} $ which assign to a complex $ K _ {\mathbf . } $ the corresponding homology groups $ H _ {i} ( K _ {\mathbf . } ) $ define a homology functor.

Let $ F: {\mathcal A} \mapsto {\mathcal A} _ {1} $ be an additive covariant functor for which the left derived functors $ R _ {i} F $ ($ R _ {i} F = 0 $, $ i < 0 $) are defined (cf. Derived functor). The system $ ( R _ {i} F ) _ {i \in \mathbf Z } $ will then define a homology functor from $ {\mathcal A} $ into $ {\mathcal A} _ {1} $.

Another example of a homology functor is the hyperhomology functor.

A cohomology functor is defined in a dual manner.

References

[1]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221