" The point about homology is that it measures defects, or obstructions ... Things in homology are about measuring defects. Snake lemmas, etc, tell you how those defects are related ... The snake lemma tells you that if you were looking at n-dim holes, then you next need to take into account the n-1 dim holes - or the boundaries of the n-dim holes. "
https://www.physicsforums.com/threads/exact-sequences.164213/
spectral sequences anyone?