Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
The analogs in (∞,1)-category theory of the adjoint functor theorems in ordinary category theory.
Let be a functor between -categories. We say that satisfies the solution set condition if the -category admits a small weakly initial set for any in .
satisfies the -initial object condition if admits an h-initial object for any in .
is said to be 2-locally small if for every pair of objects, ,, of , the mapping space is locally small.
Let be a continuous functor. Suppose that is locally small and complete and is 2-locally small. Then admits a left adjoint if and only if it satisfies the solution set condition.
Let be a finitely continuous functor. Suppose that is finitely complete. Then admits a left adjoint if and only if it satisfies the -initial object condition.
See Section 3 of (NRS18).
The following result is a consequence.
Let be an (∞,1)-functor between locally presentable (∞,1)-categories then
it has a right adjoint (∞,1)-functor precisely if it preserves small colimits;
it has a left adjoint (∞,1)-functor precisely if it is an accessible (∞,1)-functor and preserves small limits.
This is HTT, cor. 5.5.2.9.
For the existence of right adjoints, we can weaken the hypotheses to merely requiring to be a locally small (infinity,1)-category. (HTT, rem. 5.5.2.10)
Section 5.5 of
Last revised on June 10, 2021 at 05:06:16. See the history of this page for a list of all contributions to it.