Chain Maps
A chain map f between two chain complexes and is a sequence of module homomorphisms for each n that commutes with the boundary operators on the two chain complexes: . Such a map sends cycles to cycles and boundaries to boundaries, and thus descends to a map on homology:.
A continuous map of topological spaces induces chain maps in both the singular and de Rham chain complexes described above (and in general for the chain complex defining any homology theory of topological spaces) and thus a continuous map induces a map on homology. Because the map induced on a composition of maps is the composition of the induced maps, these homology theories are functors from the category of topological spaces with continuous maps to the category of abelian groups with group homomorphisms.
It is worth noticing that the concept of chain map reduces to the one of boundary through the construction of the cone of a chain map.
Read more about this topic: Chain Complex
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