TITLE: Does every l.e.s. "in homology" come from a s.e.s. of complexes?
QUESTION [18 upvotes]: Given a long exact sequence of the form
$$
\dots\to A'_n \to B'_n \to C'_n
\,\xrightarrow{\omega_n}\,
A'_{n-1} \to B'_{n-1} \to C'_{n-1}\to \dots\qquad (*)
$$
is there a way to recover a short exact sequence of complexes $\mathcal A=\{A_n,\partial_n^A\}$, $\mathcal B=\{B_n,\partial_n^B\}$, $\mathcal C=\{C_n,\partial_n^C\}$ such that the sequence (*) "is" the long exact sequence in homology induced by
$$
0\to \mathcal A\to  \mathcal B\to  \mathcal C \to 0
$$
and the morphisms $\omega_n$ are in fact the connection morphisms of that homology? I mean $A'_n\cong H_n(\mathcal A)$ for all $n\ge 0$ and similarly for $B'_n$, $C'_n$.
I expect the answer will be "obviously no", but then is there a case in which it is possible?

REPLY [6 votes]: This is not a direct answer, but related. This paper by Jan Stovicek addresses the question "which long exact sequences can arise from the snake lemma" so it might be of help here.
http://arxiv.org/abs/0906.1286