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Moritz Groth put up some excellent lecture notes: https://arxiv.org/abs/1007.2925v2 If you are more categorically minded, Emily Riehl's book has a lot about quasi-categories: http://www.math.jhu.edu/~ …

I don't know that much of the $\infty$ story, so let me phrase my answer in terms of model categories. Berger-Moerdijk's Resolution of coloured operads and rectification of homotopy algebras is a gre …

Philippe Gaucher is right. This problem was solved by Julie Bergner, here. I recently asked a question that summarized some of her work on this problem. The point is that the homotopy limit of your di …

You definitely need $M$ to be combinatorial for these types of statements. I believe Lurie has shown that every accessible localization of a presentable infinity category can be expressed as a left Bo …

In (2), you linked to Mark Hovey's paper on Smith ideals, and mentioned "the commutative framework." But Hovey explicitly writes "we have not dealt with the commutative situation at all," so I don't k …

The answer to (2) is yes, by a nice result of Gaitsgory plus an easy categorical argument. To spell it out, the situation of the OP is the following, where functors going down are $ev_i$ and functors …

This is far from a complete answer, but perhaps it will help make rigorous the idea of $(\infty,n)$ as $(\infty,1)$ enriched in $(\infty,n-1)$, and thereby give you another approach to $(\infty,\infty …

The OP wrote "I was hoping to find a reference that deals with truncation in simplicial model categories." In 2022, Michael Batanin and I published a paper, Homotopy theory of algebras of substitudes …

This seems highly related to the research program of Kathryn Hess. In 2010 she wrote a paper which lays the groundwork for homotopic descent and codescent, and that's where I would start if I were you …

I'd encourage the OP to read the writings of others on this topic, before trying to write something from scratch. I attended lectures at OSU where Aaron Mazel-Gee motivated $\infty$-categories very mu …

I don't want this question to hang around forever on the "unanswered queue," so let me add an answer, even though I think the comments largely answer it. My motivation here is to advertise a few other …

(1) No, it is not true. There are examples of adjunctions between $\infty$-categories that do not come from Quillen adjunctions. More often, they come from zigzags of Quillen adjunctions, at least if …

To me, the interest in model categories stems from Quillen's observation that the tools of topology (e.g., CW approximation) can be applied in so many different settings, especially in algebra. But no …

I will give a partial answer. I note that the OP has asked a LOT of questions recently (I count 12 so far in the first 9 days of August), and many of them are good questions on which much research has …

I answer the question "where can I read the formal definition of the presentation of ∞-categories by generators and relations?" You can read about this in the Unicity paper by Barwick and Schommer-Pri …