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A possibly simpler way of proving what you are after is using marked simplicial set. Recall that marked simplicial sets are pairs $(X,S)$ where $X$ is a simplicial set and $S\subseteq X_1$ is a set o …

I'm going to do a proof assuming we are in a stable $\infty$-category (I'm pretty sure this is almost equivalent to your "sufficiently rich" situation anyway). In your case $F=j_!j^!$ and $G=i_*i^*$. …

Well, I guess I can write as an answer what I wrote as a comment. Any pullback square where $C$ is not discrete will yield a counterexample. For simplicity let $B=G=\ast$ and $C=S^1$. Then $E=H=\Omeg …

This is worked out in Higher Algebra, example 3.2.4.4. Concretely, $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})^\otimes$ is defined as follows: it is the simplicial set over $\mathrm{Fin}_\ast$ such that …

A(n ∞-)category with $G$-action is just a functor $BG\to \mathrm{Cat}_∞$. Then, if $\mathcal{C},\mathcal{D}$ are (∞-)categories with $G$-action, we can get another (∞-)category with $G$ action $\mathr …

I already answered some version of this question in this answer, but let me try to expand a bit on the concrete advantages in mathematical practice. For understanding the following you need to take on …

The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra Theorem 5.4.5.9. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Composition with the map $$\mathrm{Disk}(M) …

This is always true, even without the hypothesis of stability. In an ∞-category a fiber sequence $X\to Y\to Z$ is a pullback square $$\require{AMScd} \begin{CD} X @>>> Y\\ @VVV @VVV \\ * @>>> Z \end{C …

In fact what you need is that your ∞-category is additive (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories …

Ok, let me try to give you a proof of something that is a lot stronger than what you asked for, but which hopefully is a bit more natural. I am basically going to smother the problem under the abstrac …

I assume that with ``monoidal ∞-groupoid'' you mean an $E_1$-space. In this case the answer is yes. It is well known that $E_1$-spaces can be modeled by functors $$X:\Delta^{\mathrm{op}}\to \operatorn …

Let $\mathcal{C},\mathcal{D}$ two $\infty$-categories and $f:\mathcal{C}\to\mathcal{D}$ and $g:\mathcal{D}\to \mathcal{C}$ two functors. Recall (HTT.5.2.2.7) that a natural transformation $u:1_{\math …

The ∞-categorical analog of the fact you mention can be found in Higher Algebra, corollary 7.3.4.14: Let $\operatorname{CAlg}$ be the category of $E_\infty$-rings and $A\in \operatorname{CAlg}$. Then …

Let $G$ be a topological group and $X$ be a paracompact Hausdorff topological space. For simplicity let us assume that $G$ has the homotopy type of a CW complex, although a lot of this answer does not …

As discussed in the comments, I'm writing here the proof of the following fact: Let $\mathscr{X}_∞$ be the ∞-topos of sheaves on $\mathrm{FinTop}^{op}$ under the atomic topology (the topology wher …