Let $i\colon\mathcal W\to\mathcal C$ be the inclusion of a subcategory. Unless I'm mistaken, the 2 out of 3 axiom for $\mathcal W$ to be a category of weak equivalences can be expressed as the existence of the dotted arrow in
where $\mathrm N$ is the nerve of a category; in other words, $\mathrm N(i)$ is a fibration.
Is this idea used anywhere, maybe in the homotopy theory of model categories?