Show that the relative singular homology group is a covariant homotopy functor from the category
of pairs of topological spaces (X,A) to the category of graded abelian groups.
Thanks a ton!!
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.
$\displaystyle \cdots H_n(A) \longrightarrow H_n(X) \longrightarrow H_n(X,A)$$\displaystyle \longrightarrow H_{n-1}(A) \longrightarrow H_{n-1}(X) \cdots \longrightarrow H_0(X, A) \longrightarrow 0 $.
$\displaystyle H_n$ is a functor from the category of topological spaces to the category of (graded) abelian groups such that $\displaystyle H_n:Top \rightarrow Ab$ given by $\displaystyle X \mapsto H_n(X)$. To check it is a covariant functor, consider the topological inclusion map $\displaystyle i:A \rightarrow X$. It induces an abelian group homomorphism $\displaystyle H_n(i):H_n(A) \rightarrow H_n(X) $ (If it induces $\displaystyle H_n(X) \rightarrow H_n(A) $, then $\displaystyle H_n$ is a contravariant functor).
In a similar vein, consider a topological pair (X, A). $\displaystyle H_n$ assigns (X, A) to $\displaystyle H_n(X, A)$ and $\displaystyle fX,A) \rightarrow (Y,B)$ to $\displaystyle H_n(f):H_n(X,A) \rightarrow H_n(Y,B)$.You also need to check $\displaystyle H_n$ satisfies the functoriality (link).