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
Famous quotes containing the words chain and/or maps:
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)
“And at least you know
That maps are of time, not place, so far as the army
Happens to be concernedthe reason being,
Is one which need not delay us.”
—Henry Reed (19141986)