Homology Theory is based on the idea that if you have a sequence $A \rightarrow B \rightarrow C$ of homomorphisms $f \colon A \rightarrow B$ and $g \colon B \rightarrow C$ such that their composition $A \rightarrow C$ is zero, then you can define the homology of that sequence to be $H(A \rightarrow B \rightarrow C) :=(ker\ g)/(im\ f)$.
More generally, for homomorphisms $f \colon A \rightarrow B$ and $g \colon B \rightarrow C$ which do not necessarily compose to zero, we could define some kind of linear symmetric difference $(ker\ g + im\ f)/(ker\ g \cap im\ f)$. If $f$ and $g$ actually do compose to zero, then this symmetric difference of course equals the homology of $A \rightarrow B \rightarrow C$, since in this case $im\ f \subset ker\ g$.
Now my question: does there exist a theory which is "similar" to homology theory, but takes arbitrary homomorphisms and the linear symmetric differences just described as its basic notions, instead of complexes and their homology?