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Suppose that $\mathcal{A}$ is an additive $\infty$-category. By this I mean that $\mathcal{A}$ is pointed, semi-additive (i.e., admits biproducts, which I will call direct sums and denote by $\oplus$) …

The answer above is correct, but apparently the non-abelianness of the extension is the only obstruction. In particular, if you assume that the middle group in the extension is abelian then it will be …

Edit: The answer below only works for the case where all the $L^i$'s are finite dimensional. The statement is true. In fact, under the assumptions of the question, it is also true that ${\scr C}(L)^* …

I don't know a reference but here is a not-too-long proof. The condition that $\mathsf{D} \to \mathsf{E}$ is a cartesian fibration implies that for every $\langle n \rangle \in \mathrm{Fin}_*$ the map …

A particular case of Lurie and Hopkins' ambidexterity theory is that if $G$ is a finite group acting on a $K(n)$-local spectrum $X$ then the norm map $$ X_{hG} \to X^{hG} $$ is a $K(n)$-local equivale …

To my understanding the situation is roughly like this. Let $\mathcal{C}$ be an $\infty$-category admiting small limits and colimits and let $f: X \to Y$ be a map of spaces whose homotopy fiber is $n$ …

Let me elaborate more on the remark above. Let $k$ be a perfect field. Let $\mathrm{Field}_k$ denote the category of finite extensions of $k$, i.e., the objects of $\mathrm{Field}_k$ are fields $k'$ e …

I believe that this is a particular case of Lurie's "operadic left Kan extension". We may identify a monoidal $\infty$-category $\mathcal{C}$ with a coCartesian fibrations of $\infty$-operads $\mathca …