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If $(\mathcal{C},W)$ is a category with weak equivalences then we may naturally form its Dwyer-Kan simplicial localisation $L(\mathcal{C}, W)$. This is a simplicial category which naturally gives a simplicial enrichment of the usual 1-categorical localisation $\mathcal{C}[W^{-1}]$. I have two questions:

  1. Is the $\infty$-category associated to $(\mathcal{C}, W)$ the simplicial nerve of a fibrant replacent of $L(\mathcal{C}, W)$?

  2. Is it actually true that $L(\mathcal{C}, W)$ is a fibrant simplicial category itself, that is all hom-spaces are $\infty$-groupoids?

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1. Almost certainly yes... 2. For some reason I want to say no, but I don't know. – Dylan Wilson Mar 1 '13 at 15:16
Currently, Harvard's Thursday Seminar is about the Unicity Theorem of Barwick and Schommer-Pries: There will be a website soon with lecture notes. The third lecture was last week and this functor $L$ was discussed, as well as a number of other functors out of RelCat. I believe the commutativity of the main diagram can be used to show (1). You should check out the paper and (when it's up) the notes from that talk. – David White Mar 1 '13 at 18:13
I can say less in the way of useful things regarding (2), because I'm out of my depth, but perhaps there will be some truth in this: you can compare $Cat_S$ (simplicial categories) with $Seg_c$ (Segal categories) where fibrant objects are Reedy fibrant diagrams. Perhaps by understanding this part of the unicity diagram you can conclude that the hom-spaces land in maps of diagrams which satisfy the necessary fibrancy. After all, if you get a Kan complex $X$ out of this then you'd know $ho(X)_\bullet = \pi_{\leq i}X$ was a groupoid, which might help in proving $X$ came from an $\infty$-groupoid. – David White Mar 1 '13 at 18:31

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