Remark There is a partial converse to Proposition If $f: X \rightarrow Y$ is a morphism between simply-connected simplicial sets and the induced map $\mathrm{H}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(Y; \operatorname{\mathbf{Z}})$ is an isomorphism, one can show that $f$ is a weak homotopy equivalence. Beware that this is not necessarily true if $X$ and $Y$ are not simply connected (see ยง for further discussion).