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adjoint (infinity,1)-functor theorem

Idea

The analog in (∞,1)-category theory of the/a adjoint functor theorem in ordinary category theory.

Statement

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.

Related concepts

• adjoint functor theorem

• adjoint $(\mathrm{\infty },1)$-functor theorem

References

Section 5.5 of

• Jacob Lurie, Higher Topos Theory