Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
An adjoint triple $F \dashv G \dashv H$ is called an ambidextrous adjunction (or sometimes ambiadjunction or ambijunction, for short) if the left adjoint $F$ and the right adjoint $H$ of $G$ are equivalent $F \simeq H$, or more precisely: equipped with a specified equivalence.
In fact, often $F$ is identified with $H$, which is the situation of a strongly adjoint pair $(F \dashv G \dashv F)$ originally considered by Morita 1965. Some authors refer to this situation by saying that $G$ is a Frobenius functor (ie. a functor which has a left adjoint that is also a right adjoint).
Sometimes $F$ is said to be biadjoint to $G$ (not to be confused with biadjoint in the sense of biadjunction). Functor $G$ which has a left and right adjoint which are equivalent is said to be Frobenius functor.
In the special case that $G$ is a fully faithful functor with an ambidextrous adjoint one also speaks of an essential localization (cf. bireflective subcategory).
The monad induced by an ambidextrous adjunction is a Frobenius monoid object in endofunctors. (e.g. Lauda 05, theorem 17), hence a Frobenius monad.
Let $\mathcal{D} \in Cat_\infty$ be an (∞,1)-category with small (∞,1)-colimits. For $f \;\colon\; X \longrightarrow Y$ a morphism of ∞-groupoids, write
for the induced pullback of (∞,1)-functor (∞,1)-categories (which one may think of as the categories of $\mathcal{D}$-valued local systems over $X$ and $Y$, respectively). The left adjoint and right adjoint (if it exists) of this are left and right (∞,1)-Kan extension.
Say that a morphism $f$ is $\mathcal{D}$-ambidextrous if $(f_! \dashv f^\ast)$ is an ambidextrous adjunction $(f_! \simeq f_\ast)$ and in addtion all pullbacks of $f$ satisfy some property (…).
Say that an ∞-groupoid $A \in Grpd_\infty$ is $\mathcal{D}$-ambidextrous if its terminal map is.
(Hopkins-Lurie 14, def. 4.1.11)
A morphism $f \colon X \to Y$ between ∞-groupoids, is $\mathcal{D}$-ambidextrous, def. , precisely if each homotopy fiber $X_y$ of $f$ is.
(Hopkins-Lurie 14, prop. 4.3.5)
(coincident limits and colimits)
Let $\mathcal{C}$ be a small category and $\mathcal{D}$ any category and consider the functor $const \mathcal{D} \longrightarrow [\mathcal{C}^{op}, \mathcal{D}]$ 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 $\mathcal{D}$ over diagrams of shape $\mathcal{C}$ coincide with the colimits over these diagrams. If $\mathcal{C}$ 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 $\mathcal{C}$ is called $\mathcal{D}$-ambidextrous (or rather, they consider $\mathcal{C}$ an ∞-groupoid and hence call it a $\mathcal{D}$-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:
The terminology “Frobenius functors” for “strongly adjoint pairs” is due to
Stefaan Caenepeel, Gigel Militaru, Shenglin Zhu, Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties, Trans. Amer. Math. Soc. 349 (1997) 4311-4342 [1997-349-11/S0002-9947-97-02004-7, pdf]
F. Castaño Iglesias, José Gómez Torrecillas, C. Nastasescu, Frobenius functors: applications, Comm. Alg. 27 10 (1998) 4879-4900 [doi:10.1080/00927879908826735]
The case of bireflective subcategories:
On the Frobenius monads induced by ambidextrous adjuntions:
Ross Street, Frobenius monads and pseudomonoids, J. Math. Phys. 45 3930 (2004) [doi:10.1063/1.1788852]
Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, Theory and Applications of Categories 16 4 (2006) 84-122 [arXiv:math/0502550, tac:16-04]
See also:
with some review in:
On the issue of equipping an ambidextrous adjunction $F \dashv G \dashv H$ with a specific equivalence between $F$ and $H$:
Connection to Hopf adjunctions
Last revised on October 30, 2023 at 10:56:06. See the history of this page for a list of all contributions to it.