Editing Derived functor (section)

== Naturality ==

Derived functors and the long exact sequences are "natural" in several technical senses.

First, given a [[commutative diagram]] of the form

:<math>\begin{array}{ccccccccc} 
0&\to&A_1&\xrightarrow{f_1}&B_1&\xrightarrow{g_1}&C_1&\to&0\\ 
&&\alpha\downarrow\quad&&\beta\downarrow\quad&&\gamma\downarrow\quad&&\\ 
0&\to&A_2&\xrightarrow{f_2}&B_2&\xrightarrow{g_2}&C_2&\to&0 
\end{array}</math>

(where the rows are exact), the two resulting long exact sequences are related by commuting squares:

[[Image:two long exact sequences.png]]

Second, suppose η : ''F'' → ''G'' is a [[natural transformation]] from the left exact functor ''F'' to the left exact functor ''G''. Then natural transformations ''R<sup>i</sup>''η : ''R<sup>i</sup>F'' → ''R<sup>i</sup>G'' are induced, and indeed ''R<sup>i</sup>'' becomes a functor from the [[functor category]] of all left exact functors from '''A''' to '''B''' to the full functor category of all functors from '''A''' to '''B'''. Furthermore, this functor is compatible with the long exact sequences in the following sense: if
:<math>0\to A\xrightarrow{f}B\xrightarrow{g}C\to 0</math>
is a short exact sequence, then a commutative diagram

[[Image:two long exact sequences2.png]]

is induced.

Both of these naturalities follow from the naturality of the sequence provided by the [[snake lemma]].

Conversely, the following characterization of derived functors holds: given a family of functors ''R''<sup>''i''</sup>: '''A''' → '''B''', satisfying the above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object ''I'' of '''A''', ''R''<sup>''i''</sup>(''I'')=0 for every positive ''i'', then these functors are the right derived functors of ''R''<sup>0</sup>.