mapping cone

Given a map of chain complexes, the mapping cone is a single new complex that packages the map together with its source and target so that the homology of the cone reports on how good the map is. Borrowed from topology — where coning off a map glues a cone onto the image — the algebraic version turns the question “is this map an isomorphism on homology?” into the question “is this one complex acyclic?”

For a chain map f : A -> B, the mapping cone Cone(f) is the complex with Cone(f)_n = A_{n-1} ⊕ B_n, equipped with the differential d(a, b) = (-d_A a, f(a) + d_B b). One checks d^2 = 0 using d^2 = 0 in A and B together with f being a chain map. There is a short exact sequence of complexes 0 -> B -> Cone(f) -> A[-1] -> 0, where A[-1] is A shifted in degree, and its long exact sequence in homology has connecting map exactly H_*(f).

The payoff is clean: because the connecting homomorphism of that long exact sequence is induced by f, the cone is acyclic (all homology zero) if and only if f induces isomorphisms on all homology — that is, if and only if f is a quasi-isomorphism. So the mapping cone converts a comparison of two complexes into a single vanishing condition, and it is the algebraic engine behind triangulated and derived categories, where it produces the distinguished triangles.