A combinatorial approximation functor sSet->qCat

Question

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap \mathbf{W}$ (the mid-anodynes are properly included in the class of trivial cofibrations)) has a small generating set with accessible source and target, Quillen's small object argument allows us to replace any simplicial set by a Joyal-equivalent quasicategory (and functorially so!).

However, as is often (would it be ungentlemanly for me to say "always"?) the case with factorizations constructed using the small object argument, it is extremely difficult to say anything concrete at all about the resulting approximations, which are typically immense (as they are constructed by a transfinite recursion).

The classical model structure on simplicial sets (denoted just as $sSet$) has an extremely elegant combinatorial fibrant replacement functor due to Dan Kan, called $\mathbf{Ex}^\infty$. The $n$-simplices of $\mathbf{Ex}^\infty S$ are exactly the k-fold subdivided n-simplices of $S$ for $k\geq 0$.

This tells us a lot of concrete information about the fibrant replacement, which we simply can't get from those approximation functors arising from the small object argument. The difference: The $k$-th stage of the transfinite composition does not depend on the previous terms. This is similar to presentations of sequences by direct (is that the right word?) formulae vs recursive formulae.

Question

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Does there exist anything similar to $\mathbf{Ex}^\infty$ for quasicategories? How about for the other widely-used simplicial models for $(\infty,1)$-categories: complete Segal spaces and Segal categories?

(Incidentally, I think that there is an analogue of $\mathbf{Ex}^\infty$ for simplicial categories gotten by applying $\mathbf{Ex}^\infty$ on hom-objects. However, this is not nearly as powerful, since not every object in $sCat$ is cofibrant).

Answer 1

$\def\Cnec{{\frak C}^{\rm nec}} \def\Choc{{\frak C}^{\rm hoc}} \def\C{{\frak C}} \def\N{{\rm N}} \def\id{{\rm id}} \def\Exi{{\rm Ex}^∞} \def\sCat{{\sf sCat}} \def\sSet{{\sf sSet}_{\sf Joyal}}$ Such a fibrant replacement functor can be constructed by composing three existing constructions:

• The Dugger–Spivak functor $$\Cnec: \sSet→\sCat$$ that sends a simplicial set $X$ to the simplicial category $\Cnec(X)$ whose set of objects is $X_0$ (the set of 0-simplices of $X$) and the simplicial set of morphisms from $x∈X_0$ to $y∈X_0$ is the nerve of the category of necklaces from $x$ to $y$. Objects of this category are necklaces from $x$ to $y$, i.e., maps of simplicial sets $N_k:=Δ^{k_1}∨⋯∨Δ^{k_m}→X$ ($m≥0$, $k_i≥1$) that map the initial vertex of $Δ^{k_1}$ to $x$, the terminal vertex of $Δ^{k_m}$ to $y$, and $∨$ denotes the operation that fuses the terminal vertex of the simplex to its left with the initial vertex of the simplex to its right. Morphisms of necklaces from $N_k→X$ to $N_l→X$ are simplicial maps $N_k→N_l$ that preserve the initial and terminal vertices and make the obvious triangle commute. This functor is a Dwyer–Kan equivalence of relative categories.

• Kan's $\Exi$ functor, applied to each mapping simplicial set in a simplicial category: $$\Exi:\sCat→\sCat.$$ This is a fibrant replacement functor in the Dwyer–Kan and Bergner model structures.

• Cordier's homotopy coherent nerve functor $$\N:\sCat→\sSet.$$ This is a right Quillen equivalence.

By construction, the composition $$\N∘\Exi∘\Cnec:\sSet→\sSet$$ is a Dwyer–Kan equivalence of relative categories that lands in quasicategories. Furthermore, we have a zigzag of natural weak equivalences $$\id→\N∘\Exi∘\C←\N∘\Exi∘\Choc→\N∘\Exi∘\Cnec$$ in the model category $\sSet$, where $\Choc$ is another functor constructed by Dugger and Spivak.

This construction satisfies the stated criterion in the original post, namely:

The $k$-th stage of the transfinite composition does not depend on the previous terms.

One could also use the functor $\C$ (the left adjoint of $\N$) instead of $\Cnec$. The functor $\C$ has a concrete and explicit (but slightly more complicated) description in terms of necklaces, similar to $\Cnec$. The advantage of using $\C$ is that one has a genuine natural weak equivalence $$\id→\N∘\Exi∘\C,$$ as opposed to a zigzag of natural weak equivalences.