nLab projection formula

Idea

Given an adjoint pair of functors $(f^\ast \dashv f_\ast)$ or $(f_! \dashv f^\ast)$ between two monoidal categories such that $f^\ast$ is a strong monoidal functor, then the projection morphism is the canonical natural transformation of the form

$$B \otimes f_\ast A \longrightarrow f_\ast (f^\ast B \otimes A)$$

or

$$f_! (f^\ast B \otimes A) \longrightarrow B \otimes f_! A$$

respectively. If these morphisms are equivalences then one often calls them the projection formula or the reciprocity relation.

Examples

Examples include the six operations setup in Grothendieck context and Wirthmüller context, respectively (in a transfer context both pushforward maps satisfy their projection formula). In particular in representation theory and in formal logic reciprocity is also called Frobenius reciprocity, see there for more.

For more examples see also at MO Where do all the projection formulas come from?

Related concepts

• monoidal adjunction

• dependent linear type theory

• Frobenius reciprocity, Beck-Chevalley condition

References

Discussion in a context of formal logic:

• Bill Lawvere, p. 6 in: Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970) 1-14 [pdf]

Discussion in real cohomology for integration of differential forms:

• Raoul Bott, Loring Tu, Prop. 6.15 in: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/978-1-4757-3951-0)

Discussion in the context of Grothendieck's yoga of six functors?:

• Martin Gallauer, p. 11 of An introduction to six-functor formalism, lecture at The Six-Functor Formalism and Motivic Homotopy Theory, Università degli Studi di Milano (Sept. 2021) [arXiv:2112.10456, pdf]

A general abstract account is in

• H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (TAC, pdf)

For the Grothendieck context of quasicoherent sheaves in E-infinity geometry the projection formula appears as remark 1.3.14 in

• Jacob Lurie, Proper Morphisms, Completions, and the Grothendieck Existence Theorem

See also

• MathOverflow, Where do all the projection formulas come from?

• MathOverflow, Ubiquity of the push-pull formula