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There is a classical "duality" between homotopy and homology groups in that homotopy groups are compatible with homotopy pullbacks (every homotopy pullback gives rise to a long exact sequence in homotopy), while homology groups are compatible with homotopy ...
Consider a push-out diagram of spaces C B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference ...