Homology functor
A functor on an Abelian category that defines a certain homological structure on it. A system
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 |
Homology functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_functor&oldid=52353