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).
