nLab
adjoint (infinity,1)-functor

Idea

The notion of adjunction between two (∞,1)-functors generalizes the notion of adjoint functors from category theory to (∞,1)-category theory.

There are many equivalent definitions of the ordinary notion of adjoint functor. Some of them have more obvious generalizations to higher category theory than others. A useful one is the characterization of adjoint functors in terms of their cographs. Recall from the discussion at cograph of a functor that two functors $L:C\to D$ and $R:D\to C$ are adjoint functors precisely if the cograph of $L$ coincides with the cograph of $R$ up to the obvious reversal of arrows

$$(L⊢R)\iff (\mathrm{cograph}(L)\simeq \mathrm{cograph}(R^{\mathrm{op}}{)}^{\mathrm{op}}).$$(L \vdash R) \Leftrightarrow (cograph(L) \simeq cograph(R^{op})^{op}) \,.

Since the notion of cograph has a rather straightforward generalization to the context of (∞,1)-categories this immediately leads to a definition of adjoint (∞,1)-functors.

Definition

Let $C$ and $D$ be (∞,1)-categories. An adjunction between $C$ and $D$ is

• an (∞,1)-functor $K\to I$ to the interval category $I=\{0\to 1\}$ which is a Cartesian fibration and a coCartesian fibration

• together with equivalences $C\stackrel{\simeq }{\to }{K}_{\{0\}}$ and $D\stackrel{\simeq }{\to }{K}_{\{1\}}$.

Two (∞,1)-functors $L:C\to D$ and $R:D\to C$ are called adjoint – with $L$ left adjoint to $R$ and $R$ right adjoint to $L$ if

• there exists an adjunction $K\to I$ in the above sense

• and $K$ is the cograph of $L$ and the opposite of the cograph of $R^{\mathrm{op}}$.

Properties

A SSet-Quillen adjunction presents an adjunction of $(\mathrm{\infty }\mathrm{,1})$-functors. See there for details.

References

This is definition 5.2.2.1 in

• Jacob Lurie, Higher Topos Theory .

There the statement ”$K$ is the cograph of $L$” is phrased as ”$L$ is associated to $K$”. See the discussion at cograph of a functor for details on this.