category theory

duality

# Contents

## Idea

While a (left or right) adjoint to a functor may be understood as the best approximation (from one side or the other) of a possibly non-existent inverse, any pair of adjoint functors restricts to an equivalence of categories on subcategories. These subcategories are sometimes known as the center of the adjunction, their objects are sometimes known as the fixed points of the adjunction.

The equivalences of categories that arise from fixed points of adjunctions this way are often known as dualities. Examples include Pontrjagin duality, Gelfand duality, Stone duality, and the Isbell duality between commutative rings and affine schemes.

## Definition

If $L \colon \mathcal{D} \longrightarrow \mathcal{C}$ is left adjoint to $R \colon \mathcal{C} \longrightarrow \mathcal{D}$, then we take the fixed points of the endofunctor $F G$ to be those objects of $\mathcal{C}$ on which the counit $\epsilon$ is an isomorphism, and take the fixed points of $G F$ to be those objects of $\mathcal{D}$ on which the unit $\eta$ is an isomorphism. The triangle identities then imply that $F$ and $G$ induce an equivalence of categories between the full subcategories of fixed points of $F G$ and the fixed points of $G F$.

Revised on April 26, 2017 09:53:21 by Urs Schreiber (131.220.184.222)