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