$\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$

Question

In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-categorical enhancement $\mathcal{D}_\textrm{qc}(X)$ inherits many nice features of $\mathcal{D}(\mathsf{QCoh}(X)),$ but that it also satisfies descent, and is in fact the (Zariski/fppf) sheafification of \begin{align*} \mathsf{Sch}&\to\mathsf{qCat}^\mathsf{st}\\ X&\mapsto\mathcal{D}(\mathsf{QCoh}(X)), \end{align*} where $\mathsf{qCat}^\mathsf{st}$ denotes the $\infty$-category of stable $\infty$-categories (with morphisms being the exact functors?).

Professor Cisinski also states that for an abelian category $\mathsf{A}$ and a full abelian subcategory $\mathsf{B}$ (perhaps satisfying come unlisted conditions), the derived category $\mathsf{D}_\mathsf{B}(\mathsf{A})$ of objects in $\mathsf{D}(\mathsf{A})$ whose cohomology lies in $\mathsf{B}$ admits an $\infty$-categorical enhancement $\mathcal{D}_\mathsf{B}(\mathsf{A}).$ However, I was not able to find a reference for the construction of $\mathcal{D}_\mathsf{B}(\mathsf{A})$ in the place I thought I might find it (chapter 1, section 3 of Higher Algebra).

In a number of papers I've read which use $\infty$-categorical techniques, $\mathcal{D}_\textrm{qc}(X)$ is defined as $$ \mathcal{D}_\textrm{qc}(X) := \underset{\operatorname{Spec}(A)\in\mathsf{Aff}_{/X}}{\lim}\mathcal{D}(\mathsf{Mod}_A). $$ With this definition, it is clear that $\mathcal{D}_\textrm{qc}(X)$ is the Zariski sheafification of $X\mapsto\mathcal{D}(\mathsf{QCoh}(X)).$ However, it is not clear to me that $h\mathcal{D}_\textrm{qc}(X)\simeq\mathsf{D}_\textrm{qc}(X).$ Presumably, the general construction of $\mathcal{D}_\mathsf{B}(\mathsf{A})$ is not defined as a similar limit, so that also leaves us with showing that the general construction of $\mathcal{D}_\mathsf{B}(\mathsf{A})$ in the case of $\mathsf{B} = \mathsf{QCoh}(X)$ and $\mathsf{A} = \mathsf{Mod}_{\mathcal{O}_X}$ agrees with $\mathcal{D}_\textrm{qc}(X)$ as defined above.

So, I ask:

1. What is the general construction of the $\infty$-categorical enhancement $\mathcal{D}_\mathsf{B}(\mathsf{A})$ of $\mathsf{D}_\mathsf{B}(\mathsf{A})$ (or at least, where might I find a reference for this construction)?
2. If, as I suspect, the definition of $\mathcal{D}_\textrm{qc}(X)$ given above does not agree with this construction, how does one show that its homotopy category is indeed $\mathsf{D}_\textrm{qc}(X)$?

Answer 1

As Harrison notes in the comments, we may define $\mathcal{D}_{\mathsf{B}}(\mathsf{A})$ as the full subcategory of $\mathcal{D}(\mathsf{A})$ consisting of objects $X$ such that $\pi_0(X[n])\in\mathsf{B}$ for all $n.$ I.e., this is the full subcategory of objects such that all cohomologies lie in $\mathsf{B}.$

As for the proof that $\mathrm{h}\mathcal{D}_{\mathrm{qc}}(X)$ is the ordinary derived category of sheaves of $\mathcal{O}_X$-modules with quasi-coherent cohomology (where $X$ is an algebraic stack), I found a reference in Hall and Rydh's "Perfect complexes on algebraic stacks," as proposition 1.3. (As far as I could tell, the remark in Harrison's comment handles the result when $\mathsf{D}_{\mathrm{qc}}(X)\simeq\mathsf{D}(\mathsf{QCoh}(X))$).

The argument is summarized below for anyone else who might have use of it.

Define:

• $\mathsf{D}_{\mathrm{qc}}(X) := \mathsf{D}_{\mathsf{QCoh}(X)}(\mathsf{Mod}(\mathcal{O}_X))$ the derived category of $\mathcal{O}_X$-modules on the lisse-'etale topos of $X$ with cohomology in $\mathsf{QCoh}(X).$
• $\mathcal{D}_{\mathrm{qc}}(X) := \mathcal{D}_{\mathsf{QCoh}(X)}(\mathsf{Mod}(\mathcal{O}_X))$ the full sub-$\infty$-category of the derived $\infty$-category of the abelian category $\mathsf{Mod}(\mathcal{O}_X)$ with cohomology in $\mathsf{QCoh}(X).$
• $\mathcal{QC}\mathsf{oh}(X) := \lim_{\operatorname{Spec}A\to X}\mathcal{D}(\mathsf{Mod}_A)$ where the limit is taken over all morphisms from affine schemes to $X.$

Let $U\to X$ be a smooth cover of $X,$ and let $U_\bullet^+$ be the \v{C}ech nerve of $U\to X,$ considered as a semi-simplicial algebraic space by forgetting the degeneracies.

Then we have maps between $\infty$-categories $$\mathcal{D}_{\mathrm{qc}}(X)\xrightarrow{\alpha}\mathcal{D}_{\mathrm{qc}}(U_{\bullet,\mathrm{et}}^+)\xrightarrow{\beta}\lim_{V\in U^+_{\bullet,\mathrm{et}}} \mathcal{D}_{\mathrm{qc}}(U^+_{\bullet,\mathrm{et}}/V)\xrightarrow{\gamma}\lim_{V\in U^+_{\bullet,\mathrm{et}}} \mathcal{D}_{\mathrm{qc}}(V)\xleftarrow{\delta}\mathcal{QC}\mathsf{oh}(X),$$ where $\alpha$ is an equivalence by unbounded cohomological descent, $\beta$ is an equivalence by a previous proposition (using the fact that having quasi-coherent cohomology can be verified locally), $\gamma$ is induced by the morphism of topoi $\epsilon : U^+_{\bullet,\mathrm{et}}/V\to V_{\mathrm{et}}$ with $\epsilon^*$ and $\epsilon_*$ exact, and $\delta$ is an equivalence because $\Delta^+\subseteq\Delta$ is right cofinal, meaning that in the definition of $\mathcal{QC}\mathsf{oh}(X),$ we may take the limit over $U^+_{\bullet,\mathrm{et}}$ instead.