Quillen pairs / $\infty$-adjunctions / adjunctions of homotopy categories

Question

Some of the examples of $\infty$-categories are those arising from model categories. I would like to ask: what is the relationship between Quillen adjunctions between model categories and adjoint pairs in the $\infty$-setting?

1. More precisely, let $\mathcal{A}, \mathcal{B}$ be two simplicially enriched model categories, is it true that every adjoint pair of $\infty$-functors between $\mathcal{A},\mathcal{B}$ comes from a Quillen pair?

2. More generally, what can be said about the relationship between: i) the category of Quillen pairs between $\mathcal{A}$ and $\mathcal{B}$ ii) the $\infty$-category of $\infty$-adjoint pairs between $\mathcal{A}$ and $\mathcal{B}$ (abusing the notation), and iii) the category of adjunctions between $\operatorname{Ho}\mathcal{A}$ and $\operatorname{Ho}\mathcal{B}$ ? (where $\operatorname{Ho}\mathcal{A}$ is the localization of $\mathcal{A}$ at the weak equivalences)

3. Same question with "adjunction" replaced by "equivalence"

For example, I know that in general not every equivalence between $\operatorname{Ho}\mathcal{A}$ and $\operatorname{Ho}\mathcal{B}$ arises from a Quillen adjunction. You can feel free to add assumptions / modify the question to make it more interesting or answerable if it is not such in the present form.

Answer 1

More precisely, let A,B be two simplicially enriched model categories, is it true that every adjoint pair of ∞-functors between A,B comes from a Quillen pair?

Assuming the model categories are combinatorial (but not necessarily simplicial), this is true up to a Quillen equivalence, i.e., every ∞-adjunction is induced by composing Quillen adjunctions with Quillen equivalences.

This is essentially Theorem 1.1 in arXiv:2110.04679.

In fact, a zigzag of a simple form (a Quillen equivalence followed by a Quillen adjunction) is sufficient: replace both combinatorial model categories with model categories of simplicial presheaves using Dugger's construction, and as long as the underlying sites have sufficiently many objects, the given Quillen functor will be induced by a functor of underlying sites.

Without using zigzags, this is likely false and counterexamples should not be too difficult to construct.

In the converse direction, every Quillen adjunction induces an ∞-adjunction. (Several references can be found in the cited paper.)

In general, there is no way to lift adjunctions between homotopy categories, since too much information is discarded.

Answer 2

(1) No, it is not true. There are examples of adjunctions between $\infty$-categories that do not come from Quillen adjunctions. More often, they come from zigzags of Quillen adjunctions, at least if everything in sight is presentable.

(2) There is an embedding of the $\infty$-category of (left) Quillen functors $lQFun(A,B)$ into the $\infty$-category of left adjoint $\infty$-functors $Fun_\infty(A,B)$. But it's not an equivalence, for the reason I said above. Similarly, every $\infty$-adjunction gives rise to an adjunction of homotopy categories, but not every such adjunction comes from an $\infty$-adjunction. Explicit counterexamples are well-known (e.g., Muro has written about this).

(3) Same issues with "equivalence" everywhere. Again, in presentable settings, $\infty$-equivalences give rise to zigzags of Quillen equivalences.