Chain equivalences give an equivalence relation on the space of chain homomorphisms. Two chain complexes are chain equivalent if there are chain maps ϕ:C_*->D_* and γ:D_*->C_* such that ϕ°γ is chain homotopic to the identity on D_* and γ°ϕ is chain homotopic to the identity on C_*.

chain complex | chain homomorphism | chain homotopy | homotopy equivalence | snake lemma