nLab
projection formula
Contents
Context
Monoidal categories
monoidal categories

With symmetry
With duals for objects
With duals for morphisms
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Category theory
category theory

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Universal constructions
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Extensions
Applications
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

See also

Last revised on February 26, 2014 at 08:39:11.
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