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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 …

To answer your question I'll need to do a fairly long digression on homotopy limits and colimits. Before I delve deep into the topic let me say that there's more than one way to describe this topic, f …

Since I have already given a similar answer recently, I don't want to be branded as the "anti-triangular" guy: the formalism of triangulated categories can be useful in certain settings. That said the …

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 …

I'm not aware of anyone writing the proof down, but I think we can patch it together as an easy consequence of several facts in Lurie's Higher Topos Theory (henceforth HTT). The statement, as I under …

TL DR: That is not enough. If you let $\psi_i:G(i)\to F(i)$ be the left adjoint of $\phi_i$ you also need the condition that for every $f:i\to j$ the canonical morphism $$\psi_jG(f)\to F(f)\psi_i$$ ad …

This is true if, instead of topological spaces, you work in a convenient category of topological spaces, in the sense of Steenrod. These are the place you want to do homotopy theory in (assuming you w …

Let me add a short observation to Dylan's fantastic answer. There is indeed a more concrete construction of the symmetric monoidal structure on the $\infty$-category of spectra: it is the localized Da …

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 …

You're almost there! The problem is that, as you've surmised, the group $\mathrm{Aut}(x)$ does not capture enough of the geometric structure of $G$. But that's easily solved: For every $x\in X$ we ca …

Zariski descent tells us that $$\operatorname{QCoh}(X)=\lim_{U\subseteq X} \operatorname{QCoh}(U)$$ where $U$ ranges through all open affines and the limit is taken in the $(2,1)$-categorical sense. S …

What you are referring to are sometimes called stacks or sheaves of categories. A famously important example is the stack $\mathrm{QCoh}$ sending a scheme $U$ to the category of quasi-coherent sheaves …

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 …

A significant technical improvement has been found by J. Shah in the theory of Kan extensions. Unfortunately I do not know of an exposition that does only the classical case, but reading the proof of …

Let me try to give some intuition by examining two important examples. One should start from the definition: the suspension $ΣX$ is the universal choice of $Y$ filling of a square $$\require{AMScd} \ …