Consider the path components functor $\pi_0:Top\to Set$. Let $\mathcal{W}$ be the wide subcategory which only contains the bijections. We know that $(Top,\pi_0^{-1}(\mathcal{W}))$ is a homotopical category.What does $Ho(Top,\pi_0^{-1}(\mathcal{W}))$ look like?
Ihaven't been able to figure this out.
Sketch of an answer :
For any space $X$, there is a continuous map $\pi_0(X)^{dis}\to X$ where $A^{dis}$ is the set $A$ with the discrete topology, which sends each path component to one of its elements.
This map induces an iso on $\pi_0$.
Can you see why that implies that $\pi_0 : Top\to Set$ actually is the localization at $\pi_0^{-1}(\mathcal W)$ ?