A Mackey functor for a finite group over a ring is a pair consisting of a covariant functor and a contravariant functor from finite sets with a -group action to -modules. The two functors agree on objects, send disjoint unions to direct sums, and satisfy a pullback identity (Webb 2000). For equivariant maps and , write and for the coordinate maps from to and , respectively. Then

For subgroups and of , write and . The quotient map induces restriction from to and transfer from to . Applied to the pullback of the quotient maps from and , the identity gives the double-coset formula

where the sum uses one representative of each double coset , and is the map induced by conjugation by (Webb 2000). In each degree, group cohomology with trivial coefficients gives an example, with its usual restriction and transfer maps (Díaz and Park 2015).

Mackey functors for a fusion system are defined on its orbit category using the same restriction, transfer, and double-coset compatibility. On an isomorphism, the covariant map is the inverse of the contravariant map. The centric orbit category is the full subcategory to which the contravariant part is restricted in the Díaz-Park sharpness conjecture (Díaz and Park 2015).

See also

Centric Orbit Category, Contravariant Functor, Covariant Functor, Group Cohomology, Pullback

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References

Díaz, A. and Park, S. "Mackey Functors and Sharpness for Fusion Systems." Homology Homotopy Appl. 17, 147-164, 2015. https://doi.org/10.4310/HHA.2015.v17.n1.a7.Webb, P. "A Guide to Mackey Functors." In Handbook of Algebra, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 805-836, 2000. https://doi.org/10.1016/S1570-7954(00)80044-3.

Cite this as:

Weisstein, Eric W. "Mackey Functor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MackeyFunctor.html

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