nLab indexed adjoint functor theorem

Contents

Contents

Idea

The analog of the adjoint functor theorem for indexed categories.

Statement

Theorem

Let ๐’ฎ\mathcal{S} be a cartesian category, let โ„‚\mathbb{C} and ๐”ป\mathbb{D} be ๐’ฎ\mathcal{S}-indexed categories which are locally small and have all colimits, and suppose further that โ„‚\mathbb{C} is well-copowered and has a separating family. Then an indexed functor F:โ„‚โ†’๐”ปF: \mathbb{C} \to \mathbb{D} has an indexed right adjoint precisely iff it is cocontinuous.

This is (Johnstone, theorem B2.4.6).

References

Last revised on April 27, 2023 at 06:32:47. See the history of this page for a list of all contributions to it.