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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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