There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\cdot|)$. Does there exist a product-preserving functorial fibrant replacement for the Joyal model structure?
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asked Aug 15, 2018 at 22:48
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4$\begingroup$ At any rate, there's no fibrant replacement functor which preserves finite limits and fibrations -- if there were, then the same argument as for the Quillen model structure would show the Joyal model structure to be right proper, which it's not. $\endgroup$– Tim CampionCommented Aug 16, 2018 at 5:03
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$\begingroup$ @TimCampion Lurie seems to use the fibrant replacement to ensure that his construction of the Yoneda embedding is a simplicial functor in 5.1.3 of HTT. Am I mistaken? $\endgroup$– Harry GindiCommented Aug 16, 2018 at 11:16
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$\begingroup$ I don't know, I haven't read that bit of HTT before. I don't understand how that's relevant. $\endgroup$– Tim CampionCommented Aug 22, 2018 at 21:22
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