View source for Homology functor

<!--
h0477801.png
$#A+1 = 21 n = 0
$#C+1 = 21 : ~/encyclopedia/old_files/data/H047/H.0407780 Homology functor
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A functor on an [[Abelian category|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|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|hyperhomology functor]].

A cohomology functor is defined in a dual manner.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "Sur quelques points d'algèbre homologique"  ''Tohoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR></table>