# nLab projection formula

Contents

### Context

#### Monoidal categories

monoidal categories

category theory

# Contents

## 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?

## References

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