How to construct a quasi-category from a category with weak equivalences?

Question

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a quasi-category (a simplicial set satisfying the weak Kan extension condition). What is the direct procedure that constructs such a quasi-category from $(\mathcal{C},W)$? (I don't mind assuming that $(\mathcal{C},W)$ is part of a model structure if it simplifies things.)

I can do it indirectly. For example, given a model category, one can use the Dwyer-Kan technology to construct a simplicial category (by simplicial localization, hammock localization or whatever), apply fibrant replacement in the bergenr model structure for simplicial categories (i.e. making the mapping complexes Kan) and then take the simplicial nerve (Lurie, HTT 1.1.5). Another way is to construct a complete Segal space by Rezk's nerve construction and then take the zero row (note that this involves a fibrant replacement in the Reedy model structure). Both methods are quite complicated and I would like to know a more explicit construction. In particular, I would like to understand what are, say, the 0-, 1- and 2-simplixes of the resulting quasi-category.

Answer 1

Every step in the following procedure is explicit, if somewhat complicated:

1. Construct the hammock localisation $L^H (\mathcal{C}, \mathcal{W})$. (See [Dwyer and Kan, Calculating simplicial localizations] for details.)
2. Apply $\mathrm{Ex}^\infty$ to every hom-space of $L^H (\mathcal{C}, \mathcal{W})$; this yields a fibrant simplicially enriched category $\widehat{L^H} (\mathcal{C}, \mathcal{W})$ because $\mathrm{Ex}^\infty$ preserves finite products, and the natural weak homotopy equivalence $\mathrm{id} \Rightarrow \mathrm{Ex}^\infty$ yields a Dwyer–Kan equivalence $L^H (\mathcal{C}, \mathcal{W}) \to \widehat{L^H} (\mathcal{C}, \mathcal{W})$. (See [Kan, On c.s.s. complexes] for details.)
3. Take the homotopy-coherent nerve of $\widehat{L^H} (\mathcal{C}, \mathcal{W})$ to get a quasicategory $\hat{N} (\mathcal{C}, \mathcal{W})$. (See [Cordier and Porter, Vogt's theorem on categories of homotopy coherent diagrams] for details.)

Let me make a few remarks to get you started.

• The objects in $L^H (\mathcal{C}, \mathcal{W})$ are the same as the objects in $\mathcal{C}$, and the morphisms are "reduced" zigzags of morphisms in $\mathcal{C}$.
• The natural weak homotopy equivalence $X \to \mathrm{Ex}^\infty (X)$ is bijective on vertices, so the Dwyer–Kan equivalence $L^H (\mathcal{C}, \mathcal{W}) \to \widehat{L^H} (\mathcal{C}, \mathcal{W})$ is actually an isomorphism of the underlying ordinary categories.
• The vertices (resp. edges) of $\hat{N} (\mathcal{C}, \mathcal{W})$ are the objects (resp. morphisms) in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$, which are the same as the objects (resp. morphisms) in $L^H (\mathcal{C}, \mathcal{W})$.

The 2-simplices of $\hat{N} (\mathcal{C}, \mathcal{W})$ are harder to describe. Conceptually, they are homotopy-coherent commutative triangles in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$, so they involve a simplicial homotopy in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$; and by thinking about the explicit description of $\mathrm{Ex}^\infty$, the simplicial homotopies in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$ are essentially zigzags of simplicial homotopies in $L^H (\mathcal{C}, \mathcal{W})$, i.e. zigzags of "reduced hammocks of width 1".