Say stuff re: group algebras
...esp. stuff on group (co)homology;
X^G, X_G := X/G
(invariant, orbits)
...are adjoint to inclusion,
are coreflection & reflection
X^G represented in set by point,
X_G isn't
[for Ab -> G, there's a reflection b/c of abelianization,
but no coreflection b/c there isn't a "largest abelian subgroup",
as each element generates a cyclic group]
in modules, M^G and M_G are also
ker/coker x -> gx - x
...and represented by Hom(Z,M) and Z \otimes M
(which are adjoint!)
Hey, what's going on is you have a "diagonal category",
and you're forgetting one or the other.
(just like Lie groups!)
also, Z[G] is a Hopf algebra.
that's probably central
Hopf algebras have a vier-group acting on them
(b/c of x -> -x and x -> x^*; natural product structure)
EG of A -> A^t and A -> \bar A
(and A -> -A -- ack!)