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$\DeclareMathOperator\colim{colim}\newcommand\uHom{\underline{\operatorname{Hom}}}$For the second question, it sounds like you have the right idea already. For the first, one way to view the situation …

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this questi …

I have just discovered a chart comparing topologies on Sch/S, made by Pieter Belmans. It includes all the topologies discussed above, and some more I haven't even heard of. It's even interactive and …

Toen proved that $RHom$ provides the internal hom in the homotopy category of dg-categories. For what you want to be true, you need something more than this: you need to know that $RHom$ is actually …

I never found any discussion about this in HTT either, but it turns out to work exactly the same way as in SGA 4. That is, let $f^* : P(D) \to P(C)$ denote the restriction functor on presheaves, and …

The model structure on the category of dg-categories presents an $(\infty,1)$-category DGCat. This structure is essentially provided by the existence of mapping spaces (or mapping $\infty$-groupoids) …

Let $sSet$ be the category of simplicial sets with the Quillen model structure. Define a homotopy monomorphism in $sSet$ to be a morphism whose homotopy fibres are empty or weakly contractible. In a …

More generally one has the following statement: if $u : C \to D$ is a quasi-fully faithful functor of dg-categories, then the induced morphism of mapping spaces in the model category of dg-categories …

One should really only talk about $K_0$ of small categories, otherwise one runs into the difficulty explained in Matthias's comment. Assuming $\mathcal{B}$ is small, let me identify $\mathcal{A}$ wit …

In some sense, the "universal" version of this fact was proved by Blumberg-Gepner-Tabuada as Proposition 3.3 in this paper. That is, they proved the analogue for stable $\infty$-categories, which is t …

Let $\Delta^1_k$ be the $k$-linear dg-category with two objects $0$ and $1$, mapping complexes $$ Map(0,0) = [k], $$ $$ Map(0,1) = [k], $$ $$ Map(1,0) = [0], $$ $$ Map(1,1) = [k] $$ where $[k] …

The functor $\Sigma$ admits an ind-adjoint if and only if it is left exact (SGA 4, Exp. I, 8.11.4). Every short exact sequence in a semisimple abelian category splits, and $\Sigma$ commutes with fini …

In the framework of $\infty$-categories, I think it is not difficult to see that your claim for $\mathcal{A} = \Delta^1$ holds for all $\infty$-groupoids. Let $K$ be an $\infty$-groupoid (Kan complex …

This is close to the notion of orbit category appearing for example in Bernhard Keller, On triangulated orbit categories, http://arxiv.org/abs/math/0503240 and section 7 of Gonçalo Tabuada, Chow …