nLab fixed point of an adjunction

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 a 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 (see Porst-Tholen 91).

Definition

Proposition

(fixed point equivalence of an adjunction)

Let

$$\mathcal{D} \underoverset {\underset{ R }{\longrightarrow}} {\overset{ L }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}$$

be a pair of adjoint functors. Say that

1. an object $c \in \mathcal{C}$ is a fixed point of the adjunction if its adjunction unit is an isomorphism

   $$c \underoverset{\simeq}{\eta_c}{\longrightarrow} R L (c)$$

   and write

   $$\mathcal{C}_{fix} \hookrightarrow \mathcal{C}$$

   for the full subcategory on these fixed objects;

2. an object $d \in \mathcal{D}$ is a fixed point of the adjunction if its adjunction counit is an isomorphism

   $$L R(d) \underoverset{\simeq}{\epsilon_d}{\longrightarrow} d$$

   and write

   $$\mathcal{D}_{fix} \hookrightarrow \mathcal{D}$$

   for the full subcategory on these fixed objects.

Then the adjunction (co-)restrics to an adjoint equivalence on these full subcategories of fixed points:

$$\mathcal{D}_{fix} \underoverset {\underset{ R }{\longrightarrow}} {\overset{ L }{\longleftarrow}} {\phantom{A}\phantom{{}_{\bot}}\simeq_{\bot}\phantom{A}} \mathcal{C}_{fix}$$

Proof

It is sufficient to see that the functors (co-)restrict as claimed, for then the restricted adjunction unit/counit are isomorphisms by definition, and hence exhibit an adjoint equivalence.

Hence we need to show that

1. for $c \in \mathcal{C}_{fix} \hookrightarrow \mathcal{C}$ we have that $\epsilon_{L(c)}$ is an isomorphism;

2. for $d \in \mathcal{D}_{fix} \hookrightarrow \mathcal{D}$ we have that $\eta_{R(d)}$ is an isomorphism.

For the first case we claim that $L(\eta_{c})$ provides an inverse: by the triangle identity it is a right inverse, but by assumption it is itself an invertible morphism, which implies that $\epsilon_{L(c)}$ is an isomorphism.

The second claim is formally dual.

Remark

Notice the above theorem and proof are valid in any 2-category admitting inverters. In fact $\mathcal{C}_{fix}$ and $\mathcal{D}_{fix}$ can be presented, respectively, as the inverter of the unit and the counit of the adjunction. Then the proof only uses the unit/counit definition of adjunction, which is valid in every 2-category (see adjunction) and general facts about invertibility.

Properties

• The fixed point construction can be seen as a 2-adjoint. See envelope of an adjunction for details.

• If the adjunction is idempotent, then the fixed objects in $\mathcal{C}$ are precisely those of the form $G d$, and dually the fixed objects in $\mathcal{D}$ are those of the form $F c$. Indeed, this is essentially the definition of an idempotent adjunction.

• At the other extreme, it may be the case that there are no fixed points of an adjunction, and the restriction to an equivalence is between empty categories. An example is the adjunction between sets and pointed sets. The left adjoint adjoins a new distinguished point and the right adjoint forgets which point is chosen. In this case, the unit of the adjunction is never surjective and so is never an isomorphism. This can be generalized to any algebraic theory that has a non-trivial constant. These examples give adjunctions which are “maximally non-idempotent”.

Examples

Gelfand duality

Gelfand duality is (a further restriction of) the fixed point equivalence of the adjunction between compactly generated Hausdorff spaces and topological algebras over the complex numbers, given by using the complex numbers as dualizing object (Porst-Tholen 91, section 4-c).

Related concepts

• envelope of an adjunction

• nucleus of a profunctor

• idempotent adjunction

References

• Hans-E. Porst, Walter Tholen, Section 2 of: Concrete Dualities in H. Herrlich, Hans-E. Porst (eds.) Category Theory at Work, Heldermann Verlag 1991 (pdf)