nLab
self-adjoint functor

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Idea

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

Examples

  • The duality involution () op:CatCat({-})^{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 CC has biproducts, then the composite Δ n\oplus \circ \Delta_n of the (discrete nn-ary) diagonal functor Δ n\Delta_n with the (nn-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 opCF : C^{op} \to C is called self-adjoint on the left if FF opF \dashv F^{op}. In this case, we have a natural isomorphism C(FA,B)C(FB,A)C(F A, B) \cong C(F B, A). (Conversely, we may talk about functors self-adjoint on the right if C(A,FB)C(B,FA)C(A, F B) \cong C(B, F A).)

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

Related concepts

Last revised on July 26, 2021 at 15:15:22. See the history of this page for a list of all contributions to it.