nLab projection formula



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Category theory



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

Bf *Af *(f *BA) B \otimes f_\ast A \longrightarrow f_\ast (f^\ast B \otimes A)


f !(f *BA)Bf !A 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 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?


Discussion in a context of formal logic:

Discussion in real cohomology for integration of differential forms:

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

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 May 19, 2023 at 12:52:22. See the history of this page for a list of all contributions to it.