# Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category

On the abstract of a paper by Emily Riehl and Dominic Verity, it is stated that

Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence.

Where can one find a proof of this statement?

On motivation for the question, HTT 4.2.4.1 would then imply that any colimit in a quasicategory could be computed as a homotopy colimit of a diagram $$F\colon\mathbf{J}\rightarrow\mathbf{C}$$, where $$\mathbf{J}$$ and $$\mathbf{C}$$ are simplicial categories.

• (I tried (but failed) to find such a proof in HTT, the cited paper, and Categorical Homotopy Theory by Riehl.) Commented Jan 26, 2019 at 23:47
• HTT.2.2.5.1.... Commented Jan 27, 2019 at 0:36
• ...Section 15.3 of Joyal's Notes on Quasi-Categories. Commented Jan 27, 2019 at 3:21
• Doesn’t it follow from the definition of “Quillen equivalence”? Apply N to a fibrant replacement of C(S). Commented Jan 27, 2019 at 4:33
• @DylanWilson Oh, I didn't know about that characterization of Quillen equivalences! (I just knew the Quillen-adjunction-such-that-derived-functors-give-an-equivalence-of-homotopy-categories definition) Would you consider posting this as an answer so that I can accept it? Also, thank you! Commented Jan 27, 2019 at 14:56

Sorry, we should have made this more clear. One proof appears as Theorem 7.2.2 in our previous paper in this series, The comprehension construction. As suggested by others, it follows from a suitably defined Yoneda embedding (which is hard to construct in the ∞-categorical setting).

Explicitly we show that a quasi-category B is equivalent to the homotopy coherent nerve of the full simplicial subcategory of the slice category qCat/B spanned by the right fibrations B/b -> B for each vertex b in B. Because the only objects we consider here are right fibrations the hom-spaces between two such in qCat/B are automatically Kan complexes.

The hard part of this is defining the map of quasi-categories from B to the homotopy coherent nerve, which by adjunction we construct as a simplicially enriched functor indexed by the "homotopy coherent realization" of B.

• This is great! Thanks! Commented Apr 23, 2020 at 20:40

Let $$\mathbf{C}$$ be a quasicategory. Using a version of the Yoneda lemma for quasicategories, Joyal constructs, in Section 15.3 of his Notes on Quasi-Categories, a simplicial category $$\overline{\mathbf{C}}$$ such that $$\mathrm{N}_\Delta(\overline{\mathbf{C}})$$ is equivalent to $$\mathbf{C}$$.

The result is actually valid for any simplicial set.

Lemma. If

is a Quillen equivalence, then the composition $$A\xrightarrow{\eta_A}G(F(A))\xrightarrow{G\left(P_{F(A)}\right)}G(P(F(A)))$$ is a weak equivalence, where $$P(A)$$ denotes the fibrant replacement of an object $$A\in\mathscr{A}$$.

Proof. See Lemma 3.5.2 of these notes for a proof.

Now, HTT 2.2.5.1 states that $$(\mathfrak{C},\mathrm{N}_\Delta)$$ gives a Quillen equivalence:

The above lemma then implies that given a cofibrant object in $$\mathit{sSets}_\mathrm{Joyal}$$ (that is, any simplicial set) $$S_\bullet$$, we have a weak equivalence $$S_\bullet\rightarrow\mathrm{N}_\Delta\left(P(\mathfrak{C}[S_\bullet])\right).$$

In particular, if $$\mathscr{C}$$ is a quasicategory, then $$Q(\mathfrak{C}[\mathscr{C}])$$ is a simplicial category whose homotopy coherent nerve is equivalent to $$\mathscr{C}$$.