universal property for singular homology group?(or subgroup of quotient group between abelian groups) Given abelian groups and maps between them:
$$
A \xrightarrow{f} B \xrightarrow{g} C
$$
with $g \circ f =0$
what is the universal property for $kerg/imf$?
I cannot find any references discussing it. Any hint and ref is appreciated, thanks!
 A: $\DeclareMathOperator{\im}{im}\DeclareMathOperator{\Tor}{Tor}$
Well it has of cause the universal property of the quotient: every map $t:\ker g \rightarrow T$ with $t(f(a))=0$ for all $a\in A$ factors uniquely via $\ker g / \im f$. I don’t think you can find a more elaborate universal property, which is why it is unlikely to be discussed or even mentioned.
The point of this quotient $\ker g / \im f$ (which we call a homology group) is not so much that it has a universal property, but that it somehow measures the exactness of the sequence $A\rightarrow B \rightarrow C$ at $B$. If the quotient is 0 the sequence is exact, if it is a nontrivial group the sequence is not exact. The larger the group gets the worse the non-exactness issue becomes.
One of the reasons to do homological algebra is that you can encode interesting properties about objects of interest (say modules or topological spaces) in the vanishing of some homology groups (the quotient you are interested in). For example a module $M$ is flat if and only if $\Tor^1(X,M)=0$ for all modules $X$ (I will not dwell on how to define $\Tor_1$, but it is essentially a homology group). Or a CW-complex is finite, if and only if all homology groups of the cellular complex vanish for almost all degrees (to be fair this is by definition of the cellular complex, but it was the easiest thing that came to my mind).
