nLab adjoint (infinity,1)-functor theorem

Idea

The analogs in (∞,1)-category theory of the adjoint functor theorems in ordinary category theory.

Statement

Definition

• Let $G:\mathcal{D} \to \mathcal{C}$ be a functor between $(\infty, 1)$-categories. We say that $G$ satisfies the solution set condition if the $(\infty, 1)$-category $c/G$ admits a small weakly initial set for any $c$ in $\mathcal{C}$.

• $G$ satisfies the $h$-initial object condition if $c/G$ admits an h-initial object for any $c$ in $\mathcal{C}$.

• $\mathcal{C}$ is said to be 2-locally small if for every pair of objects, $x$,$y$, of $\mathcal{C}$, the mapping space $map_{\mathcal{C}}(x,y)$ is locally small.

Theorem

• Let $G:\mathcal{D} \to \mathcal{C}$ be a continuous functor. Suppose that $\mathcal{D}$ is locally small and complete and $\mathcal{C}$ is 2-locally small. Then $G$ admits a left adjoint if and only if it satisfies the solution set condition.

• Let $G:\mathcal{D} \to \mathcal{C}$ be a finitely continuous functor. Suppose that $\mathcal{D}$ is finitely complete. Then $G$ admits a left adjoint if and only if it satisfies the $h$-initial object condition.

Proof

See Section 3 of (NRS18).

The following result is a consequence.

Theorem

Let $F : C \to D$ be an (∞,1)-functor between locally presentable (∞,1)-categories then

1. it has a right adjoint (∞,1)-functor precisely if it preserves small colimits;

2. it has a left adjoint (∞,1)-functor precisely if it is an accessible (∞,1)-functor and preserves small limits.

Proof

This is HTT, cor. 5.5.2.9.

Remark

For the existence of right adjoints, we can weaken the hypotheses to merely requiring $D$ to be a locally small (infinity,1)-category. (HTT, rem. 5.5.2.10)

Related concepts

• adjoint functor theorem

References

• Hoang Kim Nguyen, George Raptis, Christoph Schrade, Adjoint functor theorems for ∞-categories, Journal of the London Mathematical Society 101 2 (2019) 659-681 (arXiv:1803.01664)

Section 5.5 of

• Jacob Lurie, Higher Topos Theory