Let X be a topological space and A be a subspace of X.

For a pair (X, A), we can consider the following infinite sequence of homology groups and their homomorphisms.

.

is a functor from the category of topological spaces to the category of (graded) abelian groups such that given by . To check it is a covariant functor, consider the topological inclusion map . It induces an abelian group homomorphism (If it induces , then is a contravariant functor).

In a similar vein, consider a topological pair (X, A). assigns (X, A) to and X,A) \rightarrow (Y,B)" alt="fX,A) \rightarrow (Y,B)" /> to .You also need to check satisfies the functoriality (link).