If $A \rightarrow X$ is a cofibration of topological spaces then the induced map $H_i(X,A)\rightarrow H_i(X/A,*)$ is an isomorphism. I am aware of many proofs of this fact but none use methods from category theory? I think this should be possible because the result also holds for the case of simplicial sets, if $B \rightarrow Y$ is a cofibration of simplicial sets (equivalently, a monomorphism), then $H_i(Y,B) \rightarrow H_i(Y/B,*)$ is an isomorphism
My question is then if we can prove this fact about singular homology using the assumptions that Serre fibrations, weak equivalences and cofibrations make $\textbf{Top}$ into a model category and using the fact that for the Reedy model structure on $\textbf{Top}^ \Delta$, the cosimplicial object $U:\Delta \rightarrow \textbf{Top}:[n] \mapsto \Delta^n$ is cofibrant and some additional assumptions?