Consider a sequence $$ 0\to V_0\xrightarrow{A_0} V_1\xrightarrow{A_1} V_2 \rightarrow ...\xrightarrow{A_{n-2}}V_{n-1}\xrightarrow{A_{n-1}} V_n \to 0 $$ of linear maps $A_k$ between some vector spaces $V_k$. If $A_k\circ A_{k-1}=0$ for all $k$, one can define a cohomology $$ H^k:=\frac{\ker (A_k)}{\mathrm{im}(A_{k-1})} $$ for all $k$. Now, if instead we don't have $A_k\circ A_{k-1}=0$, then we don't have $\mathrm{im}(A_{k-1})<\ker (A_k)$ and cannot define such a quotient. But we still have $$ \mathrm{im}(A_{k-1}) < \{v\in V_k | A_k(v)\in \mathrm{im}(A_k\circ A_{k-1})\} $$ and hence can define the quotient : $$ \frac{\{v\in V_k | A_k(v)\in \mathrm{im}(A_k\circ A_{k-1})\}}{\mathrm{im}(A_{k-1})} $$ which, when $A_k\circ A_{k-1}=0$ gives back $H^k$. So this quotient seems to generalize cohomology in a certain sense.
Question : Is there a name for such a quotient ? Is it a well known concept in homological algebra ?
Remark : This question was motivated by this question.
I haven't seen this construction before, but, there is a simple way to turn it into ordinary cohomology: define $W_k=V_k/\operatorname{im}(A_{k-1}\circ A_{k-2})$. Then each $A_k$ induces a map $B_k:W_k\to W_{k+1}$ and $B_k\circ B_{k-1}=0$, so you get a chain complex. There is then an isomorphism between the cohomology of this chain complex and your "cohomology".
Another way to think about this is that the forgetful functor from the category of chain complexes to the category of arbitrary sequences has a left adjoint (namely, the operation that turns $V_k$ into $W_k$ as above). So, your "cohomology" is just "freely turn the sequence into a chain complex, and then take cohomology".