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I have been learning quasicategory theory in an attempt to understand higher topoi, and I have been trying to look at as many different sources as possible. Higher Topos Theory provides a higher categorical version of the Yoneda embedding, but it requires leaving the Joyal model structure on simplicial sets and as far as I understand requires a difficult path through many different model structures through the process of (un)straightening.

I know that Cisinski has recently published a book in which he develops the Yoneda embedding internal to the Joyal model structure. So here is my question: can chapter 6 of HTT be understood internally to the Joyal model structure using Cisinski's methods? Are there any difficulties other than the Yoneda embedding present in that portion of the theory that require leaving the Joyal model structure? Is it even worth it to learn in this way, or should I just bite the bullet and understand (un)straightening?

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    $\begingroup$ Not an answer, but if you're willing to skip some details, you can get pretty far by taking straightening/unstraightening as a black box at first. (This is sort of what I did, but it's too early to report whether this should be considered successful.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 28, 2021 at 14:14
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    $\begingroup$ Straightening/Unstraightening is unavoidable. In my book, I do it for left fibrations and avoid the version for cocartesian fibrations to introduce adjunctions. This is sufficient to study Kan extensions as well as locally presentable $\infty$-categories and higher topoi, so that one might read HTT from there. $\endgroup$
    – D.-C. Cisinski
    Commented Jan 28, 2021 at 18:46
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    $\begingroup$ However, Straightening/Unstraightening in the general case is a central feature of (higher) category theory. The fact that it is formulated through simpliccial categories or not is secondary though. We just need to make precise that there is a coherent correspondence between cocartesian fibrations with small fibers and functors with values in the $\infty$-category of small $\infty$-categories. $\endgroup$
    – D.-C. Cisinski
    Commented Jan 28, 2021 at 18:49
  • $\begingroup$ Ah, this is very clarifying, thank you! $\endgroup$
    – Emilio Minichiello
    Commented Jan 28, 2021 at 19:17

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