# Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem

Let $$S$$ be a simplicial set, $$\mathcal{C}$$ a simplicial category, and $$\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$$ a simplicial functor. The straightening and unstraigntening functors determine a Quillen adjunction $$St_{\phi} : (Set_{\Delta})_{/S} \leftrightarrows Set_{\Delta}^{\mathcal{C}} :Un_{\phi}$$ where $$(Set_{\Delta})_{/S}$$ is endowed with the contravariant model structure and $$Set_{\Delta}^{\mathcal{C}}$$ with the projective model structure. [...]

In then says that the proof is easy, but I can't manage to show that $$St_{\phi}$$ sends cofibrations to projective cofibrations. I thought that since the the class of morphisms which are sent to projective cofibrations is weakly saturated it is enough to show the result for all inclusions $$\partial \Delta^n \subseteq \Delta^n$$.

I did not have much success for the simplicial category $$\mathcal{C}$$ and the map $$\phi$$ could be anything and I have a hard time dealing with it.

Furthermore there is something else which troubles me: the model structure on the $$Set^{\mathcal{C}}_{\Delta}$$ makes no use of the simplicial enrichement on both $$\mathcal{C}$$ and $$sSet$$ so I was wondering if I was not missing something by believing that the that model structure on $$Set^{\mathcal{C}}_{\Delta}$$ is really the projective model structure coming from the Kan model structure on $$sSet$$ and not the one coming somehow from an other model stucture using the simplicial enrichement.

• I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf – Kevin Arlin Nov 25 '18 at 17:36
• Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be. – Oscar P. Nov 26 '18 at 10:25
• You might find someone to answer this at MO. – Kevin Arlin Nov 26 '18 at 18:59
• Cross-posted: mathoverflow.net/questions/316393 – Watson Nov 28 '18 at 10:37