ScalingStacks

Construction 15.1. Suppose $\mathcal{A}$ a category equipped with a subcategory $w\mathcal{A}$ that contains all the objects of $\mathcal{A}$ (i.e., a relative category in the terminology of [6]). We call the morphisms of $w\mathcal{A}$ weak equivalences. In this situation, one may form the hammock localization $\mathrm{L}^{\!\!\mathrm{H}}\mathcal{A}$ of Dwyer–Kan [16]; this is a simplicial category. One may apply to each mapping space a fibrant replacement $R$ that preserves products (e.g., $\mathrm{Ex}^{\infty}$) to obtain a category enriched in Kan complexes, which we shall denote $\mathrm{L}^{\!\!\mathrm{H}}_{f}\mathcal{A}$. We may now apply the simplicial nerve construction [28, 1.1.5.5] to obtain a $\infty$-category $\mathrm{N}\mathrm{L}^{\!\!\mathrm{H}}_{f}\mathcal{A}$, which we shall denote simply by $\mathrm{N}^{\mathrm{H}}\mathcal{A}$. We shall call $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ the $\infty$-category underlying the relative category $\mathcal{A}$.

When $\mathcal{A}$ is a simplicial model category, the simplicial localization $\mathrm{L}^{\!\!\mathrm{H}}\mathcal{A}$ is equivalent [17] to the full sub-simplicial category $\mathcal{A}^{\circ}$ spanned by the cofirant-fibrant objects. In this case, our $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ is equivalent to $\mathrm{N}\mathcal{A}^{\circ}$, as used by Lurie [28, A.2].