(also nonabelian homological algebra)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Let $\mathcal{A}$ be an abelian category and write $Ch_\bullet(\mathcal{A})$ for its category of chain complexes. Under forming chain homology
in some (any) fixed degree, a homotopy fiber sequence in $Ch_\bullet(\mathcal{A})$ is sent to a long exact sequence in $\mathcal{A}$. This is the homology long exact sequence.
Often this is considered specifically for the case that the fiber sequence in $Ch_\bullet(\mathcal{A})$ is that induced from a short exact sequence in $\mathcal{A}$. In this case the further map (that which makes the sequence “long”) is called the connecting homomorphism.
(…)
For the moment see still at fiber sequence, for instance the section long exact sequence in cohomology there.
We discuss the relation of homology long exact sequences to homotopy cofiber sequences of chain complexes. Technical details corresponding to the following survey are at mapping cone in the section Mapping cone – Homology exact sequences and fiber sequences.
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While the notion of a short exact sequence of chain complexes is very useful for computations, it does not have invariant meaning if one considers chain complexes as objects in (abelian) homotopy theory, where one takes into account chain homotopies between chain maps and takes equivalence of chain complexes not to be given by isomorphism, but by quasi-isomorphism.
For if a chain map $A_\bullet \to B_\bullet$ is the degreewise kernel of a chain map $B_\bullet \to C_\bullet$, then if $\hat A_\bullet \stackrel{\simeq}{\to} A_\bullet$ is a quasi-isomorphism (for instance a projective resolution of $A_\bullet$) then of course the composite chain map $\hat A_\bullet \to B_\bullet$ is in general far from being the degreewise kernel of $C_\bullet$. Hence the notion of degreewise kernels of chain maps and hence that of short exact sequences is not meaningful in the homotopy theory of chain complexes in $\mathcal{A}$ (for instance: not in the derived category of $\mathcal{A}$).
That short exact sequences of chain complexes nevertheless play an important role in homological algebra is due to what might be called a “technical coincidence”:
If $A_\bullet \to B_\bullet \to C_\bullet$ is a short exact sequence of chain complexes, then the commuting square
is not only a pullback square in $Ch_\bullet(\mathcal{A})$, exhibiting $A_\bullet$ as the fiber of $B_\bullet \to C_\bullet$ over $0 \in C_\bullet$, it is in fact also a homotopy pullback.
This means it is universal not just among commuting such squares, but also among such squares which commute possibly only up to a chain homotopy $\phi$:
and with morphisms between such squares being maps $A_\bullet \to A'_\bullet$ correspondingly with further chain homotopies filling all diagrams in sight.
This follows from using the basic property (see at exact sequence – Definition) that in a short exact sequence $A_\bullet \to B_\bullet \to C_\bullet$ the morphism on the right is a degreewise surjection together with a basic result in the theory of model categories or in fact that of categories of fibrant objects which is discussed in detail at homotopy pullback and also at factorization lemma:
by the existence of the projective model structure on chain complexes, we may regard every chain complex as a fibrant object and every degreewise surjection as a fibration. By the basic theorem discussed at Homotopy pullback – Properties – General these are sufficient conditions for the ordinary pullback as above to produce a chain complex that represents the homotopy-correct homotopy pullback (which, beware, is defined up “weak chain homology equivalence” only, hence up to zig-zags of quasi-isomorphism).
Equivalently, we have the formally dual result, proved using instead the existence of the injective model structure on chain complexes:
If $A_\bullet \to B_\bullet \to C_\bullet$ is a short exact sequence of chain complexes, then the commuting square
is not only a pushout square in $Ch_\bullet(\mathcal{A})$, exhibiting $C_\bullet$ as the cofiber of $A_\bullet \to B_\bullet$ over $0 \in C_\bullet$, it is in fact also a homotopy pushout.
But a central difference between fibers/cofibers on the one hand and homotopy fibers/homotopy cofibers on the other is that while the (co)fiber of a (co)fiber is necessarily trivial, the homotopy (co)fiber of a homotopy (co)fiber is in general far from trivial: it is instead the looping $\Omega(-)$ or suspension $\Sigma(-)$ of the codomain/domain of the original morphism: by the pasting law for homotopy pullbacks the pasting composite of successive homotopy cofibers of a given morphism $f : A_\bullet \to B_\bullet$ looks like this:
here
$cone(f)$ is a specific representative of the homotopy cofiber of $f$ called the mapping cone of $f$, whose construction comes with an explicit chain homotopy $\phi$ as indicated, hence $cone(f)$ is homology-equivalence to $C_\bullet$ above, but is in general a “bigger” model of the homotopy cofiber;
$A[1]$ etc. is the suspension of a chain complex of $A$, hence the same chain complex but pushed up in degree by one.
This is discussed in detail at mapping cone, see the section mapping cone - for chain complexes.
In conclusion we get from every morphim of chain complexes a long homotopy cofiber sequence
And applying the chain homology functor to this yields the long exact sequence in chain homology which is traditionally said to be associated to the short exact sequence $A_\bullet \to B_\bullet \to C_\bullet$.
In conclusion this means that it is not really the passage to homology groups which “makes a short exact sequence become long”. It’s rather that passing to homology groups is a shadow of passing to chain complexes regarded up to quasi-isomorphism, and this is what makes every short exact sequence be realized as but a special presentation of a stage in a long homotopy fiber sequence.
Lecture notes include