nLab
indexed adjoint functor theorem

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

  • R. Parรฉ, D. Schumacher, Abstract families and the adjoint functor theorems, in Indexed categories and their applications, Lecture Notes in Math. vol 661 Springer (1978)

Section B2.4 in

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