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

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.

• There is no simple procedure. The question ultimately boils down to, yes, fibrant replacement of one kind or another. (You missed that step with simplicial categories.) Fibrant replacement in the Bergner model structure is not so bad, at least – $\mathrm{Ex}^{\infty}$ (or any other fibrant replacement in $\mathbf{sSet}$ that preserves finite products) is all you need to construct fibrant replacements of simplicial categories. – Zhen Lin May 26 '14 at 7:27
• A completely different approach is to take the pushout of $N (\mathcal{W}) \hookrightarrow N (\mathcal{C})$ along $N (\mathcal{W}) \to \mathrm{Ex}^{\infty} (N (\mathcal{W}))$ and then take a fibrant replacement of that in the Joyal model structure. That will be, in some sense, the result of inverting $\mathcal{W}$ in $\mathcal{C}$ as a quasicategory. But I have no idea how to show that this is equivalent to the usual constructions. – Zhen Lin May 26 '14 at 7:30
• Thanks for the correction (fixed). Is there at least a simple description of the low dimensional simplexes? – KotelKanim May 26 '14 at 9:01
• Not really. The procedures you describe are only well-defined up to weak categorical equivalence. But there is an explicit fibrant replacement in the Bergner model structure, and hammock localisation is also explicit, so in principle you could get an explicit description of that. – Zhen Lin May 26 '14 at 9:10
• Well, I meant to ask whether there is a construction for which the low dimensional simplexes can be described explicitly. – KotelKanim May 26 '14 at 12:44

## 2 Answers

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".

• This is the first construction mentioned in the question. – Omar Antolín-Camarena May 29 '14 at 19:22

There is a simple direct procedure to extract a quasicategory from a model category, see Remark 2.8 in Meier's “Model categories are fibrant relative categories”. One simply has to apply the functor $i_1^* N_ξ$, which is a nerve-like construction for relative categories.