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?

• 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?
• 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)
• 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.

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.

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?

• 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?
• 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 $\mathcal{B}$, 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)
• 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.

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?

• 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?
• 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)
• 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.

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?

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?

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 $\mathcal{B}$, 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)

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.

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?

• 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?
• 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 $\mathcal{B}$, 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)
• 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.