Let X be a topological space and $C_n(X)$ be the singular chain complex. The homology is defined to be $H_n(X)$ = $ ker \partial_n / im \partial_{n+1}$.
What happens if we take $ K_n(X) = C_n(X) / im \partial_{n+1}$ instead?
(The idea comes from comparing the fundamental group to the fundamental groupoid)