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.


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).

  • $\begingroup$ Do you have a reference for the claim in the category of simplicial categories? $\endgroup$ Feb 14 '18 at 17:07
  • $\begingroup$ @StephenNand-Lal If I remember correctly, this comes from the fact that $\operatorname{Ex}^\infty$ preserves finite products and the induced map is a Dwyer-Kan weak equivalence to a fibrant object. $\endgroup$ Feb 14 '18 at 19:57
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    $\begingroup$ The preservation of those products is easy to prove since $\mathbf{Ex}^n(A \times B)=\mathbf{Ex}^n(A)\times \mathbf{Ex}^n(B)$ for finite n, and then filtered colimits are universal in $\operatorname{sSet}$, so you get preservation of finite products. That makes it a monoidal functor, so it lifts to the enriched categories, and the induced maps are D-K equivalences (they are equivalences on Homs and bijections on objects). A simplicially enriched category is fibrant iff all of its Hom objects are Kan complexes, so we are done. $\endgroup$ Feb 14 '18 at 20:11

$\def\Cnec{{\frak C}^{\rm nec}} \def\Choc{{\frak C}^{\rm hoc}} \def\C{{\frak C}} \def\N{{\rm N}} \def\id{{\rm id}} \def\Exi{{\rm Ex}^∞} \def\sCat{{\sf sCat}} \def\sSet{{\sf sSet}_{\sf Joyal}}$ 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.

  • $\begingroup$ Although Dugger and Spivak don't say it, I believe there is a natural transformation $\mathfrak C \Rightarrow \mathfrak C^{nec}$, i.e. $1 \Rightarrow N \mathfrak C^{nec}$. It sends a simplex $\sigma \in X_n$ to the simplicial functor $\mathfrak C \Delta^n \to C^{nec} X$ which does the obvious thing on objects, sends the "free" 1-morphism $f_{ij}$ from $i$ to $j$ to the composite $\Delta^{j-i} \to \Delta^n \xrightarrow \sigma X$, sends the homotopy $f_{jk} f_{ij} \to f_{ik}$ to the obvious map $\Delta^{k-j} \vee \Delta^{j-i} \to \Delta^{k-i}$, and is otherwise determined by 2-coskeletality. $\endgroup$
    – Tim Campion
    Oct 17 '20 at 16:39
  • $\begingroup$ Assuming this is a levelwise equivalence, the zigzag through $N Ex^\infty \mathfrak C^{hoc}$ can be avoided, in favor of a direct map $1 \to N Ex^\infty \mathfrak C^{nec}$, which obviates the main advantage of using $\mathfrak C$ instead of $\mathfrak C^{nec}$. $\endgroup$
    – Tim Campion
    Oct 17 '20 at 16:40
  • $\begingroup$ I am not sure I understood the point of using $\mathfrak{C}^{nec}$ instead of $\mathfrak{C}$ in the answer to the question. Can someone explain? Thank you! $\endgroup$ Oct 18 '20 at 20:42
  • $\begingroup$ @TimCampion: It appears to me that C Δ^n→C^nec X does not preserve composition already for n=3 or n=4, so is not a simplicial functor. Also, Dugger and Spivak give an explicit description of C in terms of equivalence classes of necklaces with additional data (Corollary 4.4 of their paper), and your construction, if correct, would provide a functorial way to extract a necklace from such an equivalence class, which seems unlikely. $\endgroup$ Oct 18 '20 at 23:05
  • $\begingroup$ @ManuelRivera: C^nec has mapping simplicial sets that are nerves of categories, C does not. For C^nec, simplices are chains of necklaces, whereas for C simplices are certain equivalence classes of such chains. Thus, C^nec is a bit simpler in that it avoids equivalence classes. $\endgroup$ Oct 18 '20 at 23:08

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