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Let me consider two model categories:

  • $ \mathsf{sSet} $: the category of simplicial sets with Joyal model structure,
  • $ \mathsf{sSetCat} $: the category of simplicially enriched categories with Bergner model structure.

In Lurie's "Higher Topos Theory", he showed there is a Quillen equivalence between them. Lurie denote it as:

  • $ \operatorname{N}: \mathsf{sSetCat} \to \mathsf{sSet} $: homotopy coherent nerve,
  • $ \mathfrak{C}: \mathsf{sSet} \to \mathsf{sSetCat} $.

However, his proof is not easy to follow I think, because of so many hyperlinks.

Is there another proof on this fact, or a document which explains the outline of his proof?

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Here's a second proof that followed Lurie's rather quickly. https://arxiv.org/abs/0911.0469

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