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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 …

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} \ …

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 …

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 …

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 …

No. Let $j:\mathbb{A}^2_k\smallsetminus\{0\}\to \mathbb{A}^2_k$ be the canonical open embedding. Then the derived pushforward $Rj_*$ is fully faithful and colimit-preserving. In particular, the subc …

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 …

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 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 $\mathcal{C}$ be an $\infty$-operad. Then Lurie in Higher Algebra, section 3.3.3 constructs a family of $\infty$-operads $$\operatorname{Mod}(\mathcal{C})^\otimes\to \operatorname{Fin}_\ast \times …

This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I …

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 …

I do not know of an answer for a general triangulated category (non-topological triangulated categories are very unusual), but as soon as you ask for some more structure the thesis follows very quickl …

In the case where $\mathcal{C}$ is presentable, this is constructed in proposition 2.9 of Robalo, Marco, $K$-theory and the bridge from motives to noncommutative motives, Adv. Math. 269, 399-550 …

It is proposition 1.4 in Segal's Categories and cohomology theories (a paper I love and I strongly encourage everyone interested in homotopy theory to read).