Many constructions of chain complexes arising in algebra and geometry, including singular homology, have the following functoriality property: if two objects "X" and "Y" are connected by a map "f", then the associated chain complexes are connected by a morphism formula_28 and moreover, the composition formula_29 of maps "f": "X" → "Y" and "g": "Y" → "Z" induces the morphism formula_30 that coincides with the composition formula_31 It follows that the homology groups formula_6 are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.