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fixed point of an adjunction

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 F:π’ŸβŸΆπ’žF \colon \mathcal{D} \longrightarrow \mathcal{C} is left adjoint to G:π’žβŸΆπ’ŸG \colon \mathcal{C} \longrightarrow \mathcal{D}, then we take the fixed points of the endofunctor FGF G to be those objects of π’ž\mathcal{C} on which the counit Ο΅\epsilon is an isomorphism, and take the fixed points of GFG F to be those objects of π’Ÿ\mathcal{D} on which the unit Ξ·\eta is an isomorphism. The triangle identities then imply that FF and GG induce an equivalence of categories between the full subcategories of fixed points of FGF G and the fixed points of GFG F.

Remarks

  • If the adjunction is idempotent, then the fixed objects in π’ž\mathcal{C} are precisely those of the form GdG d, and dually the fixed objects in π’Ÿ\mathcal{D} are those of the form FcF c. Indeed, this is essentially the definition of an idempotent adjunction.

Revised on December 19, 2017 07:32:58 by Mike Shulman (76.167.222.204)