nLab Frobenius functor

Definition

Definition

(Frobenius functor)
An adjoint pair of functors

$$(F\dashv G) \;\colon\; D \longrightarrow C$$

is called a Frobenius pair if $F$ is not only a left adjoint to $G$ but also a right adjoint to $G$, hence if we have an ambidextrous adjunction, i.e. an adjoint triple of the form

$$(F \dashv G \dashv F) \;\colon\; D \longrightarrow C \,.$$

This situation was called a strongly adjoint pair by Morita 1965, but the Caenepeel, Militaru & Zhu 1997 (followed by CGN 1998) suggested to refer to such $F$ as Frobenius functors.

Remark

(terminology)
The terminology “Frobenius functor” (Def. ) is motivated by the observation (Thm. 5.1 there, see below) of Morita 1965 (who instead speaks of “strongly adjoint pairs”!) that the functor of restriction of scalars $f^\ast$ along a homomorphism $f$ of rings has coinciding left adjoint (extension of scalars) and right adjoint (coextension of scalars) iff $f$ is a ring extension which is a “Frobenius extension” in the sense of Kasch 1961.

The suggestion to therefore refer to Morita’s “strongly adjoint pairs” as “Frobenius functors” is due to Caenepeel, Militaru & Zhu 1997, seconded by CGN 1998.

Beware of the independent terminology of “Frobenius monoidal functor” which does not refer to Frobenius functors that are also monoidal. (Similarly, one should not refer to Kasch’s Frobenius extensions as “Frobenius morphisms”!)

On the other hand, the (co)monads induced by Frobenius adjunctions are the Frobenius monads, see the discussion there.

Examples

Example

Morita 1965 proved that the extension of scalars functor for a morphism of rings $f:R\to S$ is Frobenius iff the morphism $f$ itself is a Frobenius extension in the sense of (Kasch 1961), that is: ${}_R S$ is finitely generated projective and ${}_S S_R\cong Hom_R({}_R S, {}_R R)$ as $R-S$-bimodules.

This is in the spirit of the finite-dimensional duality coded e.g. in the notion of Frobenius algebra.

References

The concept originates under the name “strongly adjoint pair” 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]

in application to the notion of “Frobenius extension” due to

• Friedrich Kasch, Projektive Frobenius-Erweiterungen, Sitzungsberichte der Heidelberger Akademie der Wissenschaften (1961) [doi:10.1007/978-3-662-25130-0, pdf]

The terminology “Frobenius functor” for this situation 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]

Further discussion:

• Stefaan Caenepeel, E. De Groot, Gigel Militaru, Frobenius Functors of the second kind, Comm. Algebra 30 (2002) 5359-5391 [arXiv:math/0106109, doi:10.1081/AGB-120015657]

• Stefaan Caenepeel, Gigel Militaru, Shenglin Zhu, Frobenius and separable functors for generalized module categories and nonlinear equations, Springer Lec. Notes in Math. 1787 (2002) [gBooks]