Given a category$\mathcal{C}$, by a localizer in $\mathcal{C}$ (or on $\mathcal{C}$) is often meant to be a class$W \subset Mor(\mathcal{C})$ of morphisms in $\mathcal{C}$ such that $(\mathcal{C},W)$ is a category with weak equivalences, intended as the input datum for the (simplicial) localization of $\mathcal{C}$ at $W$. Typically there are some extra conditions imposed on what constitutes a localizer in a given context.