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I would like to realize $QCat:=(sSet)_{\text{Joyal}}$ as an $(∞,1)-category$ in the sense of Lurie (i.e. weak Kan complex). I would like to do this in the following way:

1) First since $QCat$ is a model category it lives in $RelCat$ the category of small relative categories. So apply the hammock localizaiton and get a category enriched in simplicial sets.

2) Fibrantly replace the category in the Bergner model structure on categories enriched in simplicial sets and take the homotopy coherent nerve to get an $(\infty, 1)$-category.

Question 1: Does this procedure work?

Question 2: Are there some size issues? For example will $QCat$ really be an object of $RelCat$, also if I apply the hammock localization do I get a small simplicial enriched category?

I apologize for these boneheaded questions. (Also, I posted this question on stack exchange but got no answers so I deleted it and am posting here)

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    $\begingroup$ Why do you insist on constructing the quasicategory of quasicategories in this particular fashion? After all, one can extract a quasicategory directly from the Joyal model structure as described by Lurie in Definition 3.0.0.1 of HTT. $\endgroup$ – Dmitri Pavlov May 12 '17 at 11:40
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    $\begingroup$ The classical hammock localization, as defined by Dwyer and Kan, clearly does not give a category enriched in (small) simplicial sets: zigzags of morphisms in a large category do not form a set. $\endgroup$ – Dmitri Pavlov May 12 '17 at 12:39
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    $\begingroup$ It is "essentially locally small" though- the mapping spaces are weakly equivalent to small mapping spaces. So that should be fine. $\endgroup$ – Dylan Wilson May 12 '17 at 13:26
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    $\begingroup$ The $(∞,2)$-category of small $(∞,1)$-categories is indeed not a $(∞,1)$-category. The $(∞,1)$-category in question is obtained by discarding all noninvertible 2-arrows. In a similar way you can form the maximal subgroupoid of every $(∞,1)$-category. This is called the interior of the category and it is sometimes denoted $\iota C$ or $C^\cong$. $\endgroup$ – Denis Nardin May 13 '17 at 2:01
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    $\begingroup$ There are size issues but this idea basically works. It is kind of a pain writing it formally though, and the construction using the category of marked simplicial sets is much more explicit (since that is a simplicial model category and so it is easier to extract the underlying quasicategory) $\endgroup$ – Denis Nardin May 13 '17 at 2:05

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