Does every l.e.s. "in homology" originated from a s.e.s. of facilities?

Offered a lengthy specific series of the kind $$ \dots\to A'_n \to B'_n \to C'_n \,\xrightarrow{\omega_n}\, A'_{n-1} \to B'_{n-1} \to C'_{n-1}\to \dots\qquad (*) $$ exists a means to recoup a brief specific series of facilities $\mathcal A=\{A_n,\partial_n^A\}$, $\mathcal B=\{B_n,\partial_n^B\}$, $\mathcal C=\{C_n,\partial_n^C\}$ such that the series (*) "is" the lengthy specific series in homology generated by $$ 0\to \mathcal A\to \mathcal B\to \mathcal C \to 0 $$ and also the morphisms $\omega_n$ remain in reality the link morphisms of that homology? I suggest $A'_n\cong H_n(\mathcal A)$ for all $n\ge 0$ and also in a similar way for $B'_n$, $C'_n$.

I anticipate the solution will be "obviously no", yet after that exists an instance in which it is feasible?

2019-12-02 02:47:40
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This is not a straight solution, yet relevant. This paper by Jan Stovicek resolves the inquiry "which lengthy specific series can emerge from the serpent lemma" so it could be helpful below.

2019-12-03 04:07:01