Weak homotopy equivalence

For a topological space $X$, write ${\pi }_0(X)$ for its set of path components, as defined in Paths, path-connected spaces and path components. For $x\in X$ and $n\ge 1$, use the based cubical homotopy group ${\pi }_n(X,x)$ of Higher homotopy group by based cubes. Postcomposition gives the maps on components and the homomorphisms on based groups by Higher homotopy groups are functorial and based homotopy invariant.

A continuous map $f:X\to Y$ is a weak homotopy equivalence when both of the following hold:

• The function ${\pi }_0(f):{\pi }_0(X)\to {\pi }_0(Y)$ is bijective.
• For every $x\in X$ and every integer $n\ge 1$, the homomorphism $f_{\ast }:{\pi }_n(X,x)\to {\pi }_n(Y,f(x))$ is an isomorphism.

The quantifiers include every component and every source basepoint; no representative point is chosen in each component. Degree zero is a condition on sets, not on groups. Degree one uses the possibly nonabelian fundamental group. The definition applies to arbitrary spaces without separation or CW hypotheses and uses no choice principle.

For the empty source the second condition is vacuous, but the first forces the target to be empty: every point of a nonempty target belongs to a path component. Thus the unique empty-to-empty map is a weak homotopy equivalence, whereas an empty-to-nonempty map is not. The identity of any space, including a singleton, satisfies both conditions since its induced maps are identities.