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).

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  • $\begingroup$ Do you have a reference for the claim in the category of simplicial categories? $\endgroup$ – Stephen Nand-Lal 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. $\endgroup$ – Harry Gindi 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. $\endgroup$ – Harry Gindi Feb 14 '18 at 20:11

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