nLab ambidextrous adjunction




An adjoint triple FGHF \dashv G \dashv H is called an ambidextrous adjunction (or sometimes ambiadjunction or ambijunction, for short) if the left adjoint FF and the right adjoint HH of GG are equivalent FHF \simeq H, or more precisely: equipped with a specified equivalence.

In fact, often FF is identified with HH, which is the situation of a strongly adjoint pair (FGF)(F \dashv G \dashv F) originally considered by Morita 1965. Some authors refer to this situation by saying that GG is a Frobenius functor (ie. a functor which has a left adjoint that is also a right adjoint).

Sometimes FF is said to be biadjoint to GG (not to be confused with biadjoint in the sense of biadjunction). Functor GG which has a left and right adjoint which are equivalent is said to be Frobenius functor.

In the special case that GG is a fully faithful functor with an ambidextrous adjoint one also speaks of an essential localization (cf. bireflective subcategory).


Frobenius algebra structure

The monad induced by an ambidextrous adjunction is a Frobenius monoid object in endofunctors. (e.g. Lauda 05, theorem 17), hence a Frobenius monad.

Fiberwise characterization of ambidextrous Kan extension

Let 𝒟Cat \mathcal{D} \in Cat_\infty be an (∞,1)-category with small (∞,1)-colimits. For f:XYf \;\colon\; X \longrightarrow Y a morphism of ∞-groupoids, write

f *:[Y,𝒟][X,𝒟] f^\ast \;\colon\; [Y,\mathcal{D}] \longrightarrow [X,\mathcal{D}]

for the induced pullback of (∞,1)-functor (∞,1)-categories (which one may think of as the categories of 𝒟\mathcal{D}-valued local systems over XX and YY, respectively). The left adjoint and right adjoint (if it exists) of this are left and right (∞,1)-Kan extension.


Say that a morphism ff is 𝒟\mathcal{D}-ambidextrous if (f !f *)(f_! \dashv f^\ast) is an ambidextrous adjunction (f !f *)(f_! \simeq f_\ast) and in addtion all pullbacks of ff satisfy some property (…).

Say that an ∞-groupoid AGrpd A \in Grpd_\infty is 𝒟\mathcal{D}-ambidextrous if its terminal map is.

(Hopkins-Lurie 14, def. 4.1.11)


A morphism f:XYf \colon X \to Y between ∞-groupoids, is 𝒟\mathcal{D}-ambidextrous, def. , precisely if each homotopy fiber X yX_y of ff is.

(Hopkins-Lurie 14, prop. 4.3.5)



(coincident limits and colimits)

Let 𝒞\mathcal{C} be a small category and 𝒟\mathcal{D} any category and consider the functor const𝒟[𝒞 op,𝒟]const \mathcal{D} \longrightarrow [\mathcal{C}^{op}, \mathcal{D}] that sends objects to constant presheaves with this value. Then the right adjoint of this functor is, if it exists, the limit construction, and the left adjoint is, if it exists, the colimit construction. (See also at Kan extension.) Therefore if both exist as an ambidextrous adjunction, then this means that limits in 𝒟\mathcal{D} over diagrams of shape 𝒞\mathcal{C} coincide with the colimits over these diagrams. If 𝒞\mathcal{C} is a finite set, then this situation is traditionally referred to as biproducts. Generally therefore this is sometimes called bilimits (but see the discussion of the terminology there).

In (Hopkins-Lurie 14, section 4.3) such 𝒞\mathcal{C} is called 𝒟\mathcal{D}-ambidextrous (or rather, they consider 𝒞\mathcal{C} an ∞-groupoid and hence call it a 𝒟\mathcal{D}-ambidextrous space). Concrete examples of this include those discussed at K(n)-local stable homotopy theory.


(Yoga of six functors)

A Wirthmüller context in the presence of an un-twisted Wirthmüller isomorphism is an ambidextrous adjunction.


Every self-adjoint functor forms an ambidextrous adjunction.


Ambidextrous adjunctions were maybe first considered under the name strongly adjoint pairs (of functors), in:

  • Kiiti Morita, Adjoint pairs of functors and Frobenius extensions, Science Reports of the Tokyo Kyoiku Daigaku, Section A 9 202/208 (1965) 40-71 [jstor:43698658]

The terminology “Frobenius functors” for “strongly adjoint pairs” is due to

The case of bireflective subcategories:

On the Frobenius monads induced by ambidextrous adjuntions:

See also:

with some review in:

On the issue of equipping an ambidextrous adjunction FGHF \dashv G \dashv H with a specific equivalence between FF and HH:

Connection to Hopf adjunctions

  • Harshit Yadav, Frobenius monoidal functors from (co)Hopf adjunctions, arXiv:2209.15606

Last revised on October 30, 2023 at 10:56:06. See the history of this page for a list of all contributions to it.