Canonical comparison between $\infty$ and ordinary derived categories

Question

This question is a follow-up to a previous question I asked.

If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\mathcal{D}(\mathsf{A})$ is equivalent to the ordinary derived category of $\mathsf{A},$ $\mathsf{D}(\mathsf{A}).$ This is shown in Lurie's Higher Algebra by describing the homotopy category in question explicitly.

From what I've read on these matters, this equivalence is more natural that what I've described above. I have not been able to make sense of this naturality, however. I was hopeful that there was a canonical morphism $\mathcal{D}(\mathsf{A})\to N_{\bullet}(\mathsf{D}(\mathsf{A})),$ corresponding to a morphism $h\mathcal{D}(\mathsf{A})\to\mathsf{D}(\mathsf{A})$ by adjunction which is an equivalence. But this doesn't seem to be the case, since the comparison maps that I am aware of which relate the ordinary nerve to the differential graded nerve seem to go in the wrong direction.

Question 1: Does such a natural/functorial comparison exist, and if so, how does one construct it?

Question 2: Most likely, this will follow formally from question 1, but if $\mathcal{D}_{\mathrm{qc}}(X)$ denotes the derived $\infty$-category of quasi-coherent sheaves on an ordinary scheme $X,$ then can we obtain a canonical/functorial equivalence $h\mathcal{D}_{\mathrm{qc}}(X)\simeq\mathsf{D}_{\mathrm{qc}}(X)$ as well? What about if $X$ is an algebraic stack?

Question 3: My last question is regarding the homotopy category of a derived algebraic stack $\mathcal{X}.$ If we view $\mathcal{X}$ as a sheaf on the 'etale site of the $\infty$-category of animated rings valued in spaces, let $\mathcal{X}_{\mathrm{cl}}$ denote the higher algebraic stack obtained by restricting to the category of discrete commutative rings inside animated rings, and let $\tau_{\leq1}\mathcal{X}_{\mathrm{cl}}$ denote the classical algebraic stack obtained from $\mathcal{X}_{\mathrm{cl}}$ by truncating every $\mathcal{X}(A)$ to obtain an ordinary groupoid. Is it known whether some statement like $h\mathcal{D}_{\mathrm{qc}}(\mathcal{X})\simeq\mathsf{D}_{\mathrm{qc}}(\tau_{\leq1}\mathcal{X}_{\mathrm{cl}})$ true in a natural way, as in the previous questions?

Note: I know there's not a canonical definition of a derived algebraic stack. I would be happy for a result using any definition (as long as the definition is specified), and even happier if it held for more than one definition. In question 3 I assumed that the construction $\tau_{\leq1}\mathcal{X}_{\mathrm{cl}}$ will be an algebraic stack if $\mathcal{X}$ is a derived algebraic stack, although I don't know if this is always true for any of the common definitions of derived algebraic stack. A definition of derived algebraic stack where this is always true would be ideal.

Edit: To be more specific about what such a comparison should satisfy, there should be an equivalence $\gamma_X : h\mathcal{D}_{\mathrm{qc}}(X)\to\mathsf{D}_{\mathrm{qc}}(X)$ (or in the other direction, if that's more natural) for every [affine] scheme/algebraic stack $X,$ such that given a morphism of schemes/stacks $f : X\to Y,$ the following diagram commutes: $$\require{AMScd}\begin{CD} h\mathcal{D}_{\mathrm{qc}}(Y) @>{\mathbf{L}f^*}>> h\mathcal{D}_{\mathrm{qc}}(X)\\ @VV{\gamma_Y}V @VV{\gamma_X}V\\ \mathsf{D}_{\mathrm{qc}}(Y) @>>{\mathbf{L}f^*}> \mathsf{D}_{\mathrm{qc}}(X). \end{CD}$$

Answer 1

I learned the following partial answer from Peter Haine (any errors are of course my own). In the following I will ignore any set-theoretic issues which may arise.

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Let $\mathsf{A}$ be an abelian category, with derived category $\mathsf{D}(\mathsf{A})$ and derived $\infty$-category $\mathcal{D}(\mathsf{A}),$ and recall that $\mathsf{D}(\mathsf{A})$ is the $1$-categorical localization of $\mathsf{Ch}(\mathsf{A})$ with respect to the class $\mathsf{qis}$ of quasi-isomorphisms, and that $\mathcal{D}(\mathsf{A})$ is the $\infty$-categorical localization of $\mathsf{Ch}(\mathsf{A})$ with respect to $\mathsf{qis}.$

If $[1]$ denotes the walking morphism, and $\mathsf{I}$ is the walking isomorphism, one model for this localization is the pushout of $$\coprod_{f\in\mathsf{qis}} \mathsf{I}\leftarrow \coprod_{f\in\mathsf{qis}} [1] \to \mathsf{Ch}(\mathsf{A}),$$ formed either in the $\infty$-category $\mathcal{C}\mathsf{at}_1$ of $1$-categories (for $\mathsf{D}(\mathsf{A})$) or the $\infty$-category $\mathcal{C}\mathsf{at}_\infty$ of $\infty$-categories (for $\mathcal{D}(\mathsf{A})$), where in the latter we interpret $1$-categories as $\infty$-categories via their nerves.

If we write $\mathrm{N} : \mathcal{C}\mathsf{at}_1\to\mathcal{C}\mathsf{at}_{\infty}$ for the inclusion of $1$-categories into $\infty$-categories, then $\mathrm{N}$ has a left adjoint, the homotopy category functor $\mathrm{h}.$

We then obtain a map $$\mathcal{D}(\mathsf{A})\simeq\operatorname{colim}\left(\coprod_{f\in\mathsf{qis}} \mathrm{N}(\mathsf{I})\leftarrow \coprod_{f\in\mathsf{qis}} \mathrm{N}([1]) \to \mathrm{N}(\mathsf{Ch}(\mathsf{A}))\right)\to \mathrm{N}\left(\operatorname{colim}\left(\coprod_{f\in\mathsf{qis}} \mathsf{I}\leftarrow \coprod_{f\in\mathsf{qis}} [1] \to \mathsf{Ch}(\mathsf{A})\right)\right)\simeq\mathrm{N}\left(\mathsf{D}(\mathsf{A})\right)$$ by the universal property of the colimit, and the adjunction of $\mathrm{N}$ and $\mathrm{h}$ then provide the desired canonical morphism $$ \mathrm{h}\left(\mathcal{D}(\mathsf{A})\right)\to\mathsf{D}(\mathsf{A}). $$

We immediately see that this morphism is an equivalence, since $\mathrm{h}$ being a left adjoint implies it commutes with colimits, and $\mathrm{h}\mathrm{N}(\mathsf{C})\simeq\mathsf{C}$ (naturally!) for all $\mathsf{C}\in\mathcal{C}\mathsf{at}_1.$

Additionally, we obtain the desired result for $\mathcal{D}_{\mathrm{qc}}(X)$ (despite the fact that this is not the derived $\infty$-category of some abelian category in general) by noticing that the category $\mathcal{D}_{\mathrm{qc}}(X)\subseteq\mathcal{D}(X)$ is defined by a condition on cohomology (which $\mathrm{h}$ respects).

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In fact this construction then provides a functor from ``relative $\infty$-categories'' (i.e., the $\infty$-category whose objects are pairs $(\mathcal{C},\mathcal{W}),$ where $\mathcal{C}$ is an $\infty$-category and $\mathcal{W}$ is a class of morphisms in $\mathcal{C},$ and whose morphisms $F : (\mathcal{C},\mathcal{W})\to(\mathcal{C}',\mathcal{W}')$ are functors of $\infty$-categories $F : \mathcal{C}\to\mathcal{C}'$ such that $F(\mathcal{W})\subseteq\mathcal{W}'$) to the $\infty$-category of $\infty$-categories.

The desired property that these functorial equivalences should be compatible with derived pullback of sheaves should follow from expressing the derived pullback as an appropriate functor of relative $\infty$-categories, and noting that there is functoriality of these pushouts, but I have to think about this aspect of the construction more.

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If I've made any errors in my recollection of the argument I learned, or if there are technicalities or subtleties I'm not doing justice, please do let me know. I'll leave the question open for now since the question of compatibility with derived pullbacks hasn't been completely answered yet.