nLab indexed adjoint functor theorem

Contents

Context

Category theory

Contents

Idea
Statement
References

Idea

The analog of the adjoint functor theorem for indexed categories.

Statement

Theorem

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

This is (Johnstone, theorem B2.4.6).

References

Robert Paré, Dietmar Schumacher, Abstract families and the adjoint functor theorems, in Indexed categories and their applications, Lecture Notes in Math. 661 Springer (1978) [doi:10.1007/BFb0061361]
Peter Johnstone, Section B2.4 in: Sketches of an Elephant

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