Editing Homological algebra (section)

== Functoriality ==
A [[continuous map]] of topological spaces gives rise to a homomorphism between their ''n''th [[homology group]]s for all ''n''. This basic fact of [[algebraic topology]] finds a natural explanation through certain properties of chain complexes. Since it is very common to study
several topological spaces simultaneously, in homological algebra one is led to simultaneous consideration of multiple chain complexes.

A '''morphism''' between two chain complexes, <math> F: C_\bullet\to D_\bullet,</math> is a family of homomorphisms of abelian groups <math>F_n: C_n \to D_n</math> that commute with the differentials, in the sense that <math>F_{n-1} \circ d_n^C = d_n^D \circ F_n</math> for all ''n''. A morphism of chain complexes induces a morphism <math> H_\bullet(F)</math> of their homology groups, consisting of the homomorphisms <math>H_n(F) : H_n(C) \to H_n(D)</math> for all ''n''. A morphism ''F'' is called a '''[[quasi-isomorphism]]''' if it induces an isomorphism on the ''n''th homology for all ''n''.

Many constructions of chain complexes arising in algebra and geometry, including [[singular homology]], have the following [[functor]]iality property: if two objects ''X'' and ''Y'' are connected by a map ''f'', then the associated chain complexes are connected by a morphism <math>F=C(f) : C_\bullet(X) \to C_\bullet(Y),</math> and moreover, the composition <math>g\circ f</math> of maps ''f'':&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' and  ''g'':&nbsp;''Y''&nbsp;&rarr;&nbsp;''Z'' induces the morphism <math>C(g\circ f): C_\bullet(X) \to C_\bullet(Z)</math> that coincides with the composition <math>C(g) \circ C(f).</math> It follows that the homology groups <math>H_\bullet(C)</math> are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.

The following definition arises from a typical situation in algebra and topology. A triple consisting of three chain complexes <math>L_\bullet, M_\bullet, N_\bullet</math> and two morphisms between them, <math>f:L_\bullet\to M_\bullet, g: M_\bullet\to N_\bullet,</math> is called an '''exact triple''', or a '''short exact sequence of complexes''', and written as

: <math> 0 \longrightarrow L_\bullet \overset{f}{\longrightarrow} M_\bullet \overset{g}{\longrightarrow} N_\bullet \longrightarrow 0,</math>

if for any ''n'', the sequence

: <math> 0 \longrightarrow L_n \overset{f_n}{\longrightarrow} M_n \overset{g_n}{\longrightarrow}
N_n \longrightarrow 0 </math>

is a [[short exact sequence]] of abelian groups. By definition, this means that ''f''<sub>''n''</sub> is an [[injection (mathematics)|injection]], ''g''<sub>''n''</sub> is a [[surjection]], and Im ''f''<sub>''n''</sub>&nbsp;=&nbsp; Ker ''g''<sub>''n''</sub>. One of the most basic theorems of homological algebra, sometimes known as the [[zig-zag lemma]], states that, in this case, there is a '''long exact sequence in homology'''

: <math> \cdots \longrightarrow H_n(L) \overset{H_n(f)}{\longrightarrow} H_n(M) \overset{H_n(g)}{\longrightarrow} H_n(N) \overset{\delta_n}{\longrightarrow} H_{n-1}(L) \overset{H_{n-1}(f)}{\longrightarrow} H_{n-1}(M) \longrightarrow \cdots, </math>

where the homology groups of ''L'', ''M'', and ''N'' cyclically follow each other, and ''&delta;''<sub>''n''</sub> are certain homomorphisms determined by ''f'' and ''g'', called the '''[[connecting homomorphism]]s'''.  Topological manifestations of this theorem include the [[Mayer–Vietoris sequence]] and the long exact sequence for [[relative homology]].
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Functionality of ''homology'' is so fundamental a property that homological algebra takes it for granted. On the other hand, ''chain complexes'' may or may not be functorial in their 'arguments', the objects whose inner structure they are supposed to reflect. This difference serves as a source of constant tension in homological algebra. On the one hand, homology yields invariants of [[de Rham cohomology|smooth manifolds]], [[group cohomology|groups]], [[Hochschild homology|algebras]], and so on; and these invariants have extra algebraic structure, such as multiplication in the cohomology of manifolds, or the [[Mayer-Vietoris sequence]] relating homology of ''U'', ''V'', their union and their intersection. On the other hand, to ''describe'' or even ''define'' this structure, one must inevitably work with the complexes themselves. In particular, many constructions in homological algebra, such as [[connecting homomorphism]], involve intermediate non-canonical choices, which, however, do not affect the final outcome on the level of homology.
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