nLab self-adjoint functor

Idea

A self-adjoint functor is an endofunctor $F : C \to C$ such that $F \dashv F$, so that $F$ is both left adjoint and right adjoint to itself.

Examples

• The duality involution $({-})^{op} : Cat \to Cat$ is self-adjoint. More generally, this is true of the underlying category of any 2-category with a duality involution.
• If a category $C$ has biproducts, then the composite $\oplus \circ \Delta_n$ of the (discrete $n$-ary) diagonal functor $\Delta_n$ with the ($n$-ary) biproduct functor $\oplus$ is self-adjoint.

Functors self-adjoint on the left

There is a similar phenomenon involving a change of variance. A functor $F : C^{op} \to C$ is called self-adjoint on the left if $F \dashv F^{op}$. In this case, we have a natural isomorphism $C(F A, B) \cong C(F B, A)$. (Conversely, we may talk about functors self-adjoint on the right if $C(A, F B) \cong C(B, F A)$.)

• The contravariant powerset functor $\mathcal{P}: Set \to Set^{op}$ is left-adjoint to $\mathcal{P}^{op} : Set^{op} \to Set$, i.e. self-adjoint on the right.
• More generally, in a symmetric monoidal closed category $(C, \otimes, I, \multimap)$, for a fixed object $A$, the functor $(-) \multimap A$ is self-adjoint on the right.

Related concepts

• adjoint triple

• ambidextrous adjunction