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Proposition 4.20 in the same paper shows that the localization is in fact $\omega$-accessible, i.e. the fully faithful right adjoint in question preserves filtered colimits. This implies that the loca …

First let me point out a small typo : it should be functors over $\mathcal C$ which send cocartesian edges to cocartesian edges (this doesn't matter for left fibrations, but for cocartesian fibrations …

The answer is yes, but $\mathrm{LFib}(\mathcal C)$ is also the full subcategory of $(\mathrm{Cat}_\infty)_{/\mathcal C}$, it just so happens that you can prove that any morphism between such is a left …

Let me try to expand on what I wrote in the comments. I'll focus on left Kan extensions- I think the story for right Kan extensions is a bit more subtle, for reasons I'll ty to mention at the end. A ( …

There is a reference for this in Lurie's Higher topos theory - specifically, see Theorem 4.2.4.1, Corollary 4.2.4.8. The key to these is Proposition 4.2.4.4, which itself relies heavily on Appendix A …

Yes, but this is a completely general phenomenon unrelated to animated rings and sheaves. The general (surprising!) phenomenon is that the left Kan extension of any product preserving functor $C\to An …

First, a clarification about the question : as Marius points out in the comments, it sort of depends on what you mean by Lurie tensor product. If you mean the one at the level of presentable categorie …

$Idem$ is in fact compact - it is a retract of the following finite category $C$: it has a single object $x$, and is free on an the endomorphism $e$ of $x$, the homotopy $h:e^2\simeq e$ and the higher …

"This is equivalent to the assertion that the construction of the large diagram computes the suspension functor $\Sigma$" My previous answer was based on me misreading this quote :) You want to show t …

I'm going to deal with the case of $\mathcal E^1\subset \mathcal E^0$. Note that by composition of pullbacks, $\mathcal E^0 = \mathcal C_{/X}\times_{\mathcal P(S)}\mathcal P(S)_{j(s)/}$, while $\math …

You can also use Quillen's theorem A : to prove that $C \to D$ is initial, it suffices to show that for every $d\in D$, $C \times_D D_{/d}$ is weakly contractible. In the case where $C\to D$ is fully …

$E_0$ is not quite "no extra structure" : you know who $A$ is inside $Mod(A)$, so it's a pointed category (more generally, an $E_0$ object in $\mathcal S$ is an object with a "unit" $\mathbb 1\to X$). …

If by "$R$ is the unit object" you mean : 1- the object $R$ in $Mod_R$ is the unit and 2- The induced commutative algebra structure on $R = map(1,1)$ is the given commutative algebra structure on $R$, …

One has to unravel the language a little bit, but in Riehl-Verity you are looking for: Theorem 4.3.9, Theorem 4.3.11, Propositions 4.4.7, 4.4.11, 4.4.17, Theorem 4.4.18. All of them are special cases …

A reference for exactly this type of problem in general is a paper by Horev and Yanovski called "On conjugates and adjoint descent". Given a diagram $C \to D_i$ of left adjoints $f_i$ with right adjoi …