The free-forgetful adjunction between groups and sets lifts to an $\infty$-adjunction: $F \dashv U: \mathcal{S} \rightleftarrows \operatorname{Mon}_{E_1}(\mathcal{S})$ where $F$ is the free $E_1$-algebra functor and $U$ is the forgetful functor from $E_1$-algebras (associative monoids up to coherent homotopy) to spaces.

For pointed spaces, the suspension-loop adjunction $\Sigma \dashv \Omega: \mathcal{S}_* \rightleftarrows \mathcal{S}_*$ is an $\infty$-adjunction:

$\operatorname{Map}_*({\Sigma X, Y}) \simeq \operatorname{Map}_*({X, \Omega Y})$

This is the $\infty$-categorical version of the classical adjunction $[\Sigma X, Y] \cong [X, \Omega Y]$.

The nerve-fundamental category adjunction $\tau_1 \dashv N: \mathbf{sSet} \rightleftarrows \mathbf{Cat}$ lifts to an $\infty$-adjunction between the $\infty$-category of $\infty$-categories and the $\infty$-category of $1$-categories.

For any localization $L: \mathcal{C} \to \mathcal{C}_L$ (left adjoint to the inclusion $\mathcal{C}_L \hookrightarrow \mathcal{C}$ of local objects), the pair $L \dashv i$ is an $\infty$-adjunction.

Example: $n$-truncation $\tau_{\leq n}: \mathcal{S} \to \mathcal{S}_{\leq n}$ is left adjoint to the inclusion of $n$-truncated spaces.

An $\infty$-adjunction $F \dashv G$ has:

• Unit: $\eta: \mathrm{id}_{\mathcal{C}} \to G \circ F$ (a natural transformation, i.e., a $1$-simplex in $\operatorname{Fun}(\mathcal{C}, \mathcal{C})$).
• Counit: $\varepsilon: F \circ G \to \mathrm{id}_{\mathcal{D}}$.

The triangle identities hold up to homotopy:

• $(\varepsilon F) \circ (F\eta) \simeq \mathrm{id}_F$
• $(G\varepsilon) \circ (\eta G) \simeq \mathrm{id}_G$

In an $\infty$-category, "up to homotopy" means the space of witnesses for the triangle identity is contractible, not just nonempty.

In an $\infty$-category, if $F$ has a right adjoint, the right adjoint is unique up to contractible choice (unique up to a canonical equivalence). This is stronger than uniqueness up to isomorphism in ordinary categories: the space of right adjoints to $F$ is either empty or contractible.

Similarly, the unit and counit are essentially unique: the space of adjunction data for a given pair $(F, G)$ is either empty or contractible.

As in ordinary category theory:

• Left adjoints preserve colimits: if $F \dashv G$, then $F$ preserves all colimits that exist in $\mathcal{C}$.
• Right adjoints preserve limits: $G$ preserves all limits that exist in $\mathcal{D}$.

This is proved using the mapping space characterization: $\operatorname{Map}(F(\operatorname{colim} X_i), d) \simeq \operatorname{Map}(\operatorname{colim} X_i, Gd) \simeq \lim \operatorname{Map}(X_i, Gd) \simeq \lim \operatorname{Map}(FX_i, d) \simeq \operatorname{Map}(\operatorname{colim} FX_i, d)$

Every Quillen adjunction $F \dashv G: \mathcal{M} \rightleftarrows \mathcal{N}$ induces an $\infty$-adjunction $\mathbf{L}F \dashv \mathbf{R}G: \mathcal{M}_\infty \rightleftarrows \mathcal{N}_\infty$ between the underlying $\infty$-categories. This captures strictly more information than the derived adjunction on homotopy categories: the mapping space equivalence is a full equivalence of Kan complexes, not just a bijection of $\pi_0$.

For a morphism $f: X \to Y$ of schemes (or derived schemes), there is an $\infty$-adjunction:

$f^*: \operatorname{QCoh}(Y) \rightleftarrows \operatorname{QCoh}(X) : f_*$

where $f^*$ is the derived pullback and $f_*$ is the derived pushforward. These are $\infty$-categorical adjunctions between the $\infty$-categories of quasi-coherent sheaves.

Left and right Kan extensions along a functor $f: \mathcal{C} \to \mathcal{D}$ give $\infty$-adjunctions:

$f_!: \operatorname{Fun}(\mathcal{C}, \mathcal{E}) \rightleftarrows \operatorname{Fun}(\mathcal{D}, \mathcal{E}) : f^*$

$f^*: \operatorname{Fun}(\mathcal{D}, \mathcal{E}) \rightleftarrows \operatorname{Fun}(\mathcal{C}, \mathcal{E}) : f_*$

where $f_! \dashv f^* \dashv f_*$. These generalize the six-functor formalism of sheaf theory.

An $\infty$-adjunction $F \dashv G: \mathcal{C} \rightleftarrows \mathcal{D}$ is monadic if $\mathcal{D}$ is equivalent to the $\infty$-category of $T$-algebras, where $T = G \circ F$ is the associated monad. The $\infty$-categorical Barr--Beck theorem (Lurie) characterizes monadic adjunctions: $F \dashv G$ is monadic iff $G$ is conservative and preserves $G$-split simplicial colimits.