A map $f: X \to Y$ is called a **weak homotopy equivalence** if the induced maps $f_i^* : \pi_i(X, x_0) \to \pi_i(Y, f(x_0))$ are isomorphisms for every $i \geq 0$.

This is a strictly weaker notion than homotopy equivalence - for example, let $L$ be the long line. Then $\pi_i(L) = 0$ for all $i$, but $L$ is not contractible, and thus $L \not\sim \pt$. However, the inclusion $\pt \injects L$ is a weak homotopy equivalence, which can not be a homotopy equivalence.

Any weak homotopy equivalence induces isomorphisms on all integral co/homology groups, and thus co/homology groups with any coefficients by the UCT.