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It's generally best not to leave questions without an answer, even if they are answered in the comments. MO best practice is to post a CW answer summarizing the answer from the comments. In this case, …

A great reference for these types of questions is Cisinski's book Higher categories and homotopical algebra. Definition 2.2.8 on page 35 is for one-categorical localization, and Definition 7.1.2 on pa …

This is a great question. Let me start with limits and discuss lax limits later. Given a $D$-shaped diagram $X$ of model categories (where $D$ is a small category), one can ask whether the two ways (B …

In the years since this question was asked, some of the theory the OP wanted was developed. First, in Lurie's Higher Algebra, Definition 7.3.2.14 defines the absolute cotangent complex functor $L: C\t …

This week, we learned that another example is the category of simplicial sets, and the class of weak equivalences the simplicial homotopy equivalences. All credit to Tom Goodwillie, Tim Campion, and T …

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 …

Yes, this is chapter 7 of Fosco Loregian's thesis, linked from his webpage. The paper Simone Virili linked to is one of 3 papers making up the thesis. Specifically, Section 7.2.1 discusses the topolog …

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 …

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 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 …

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 …

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'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 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 …