  Let R be a (unital) commutative ring, and G be a finite group with order
invertible in R. We introduce new idempotents in the double Burnside algebra
RB(G,G), indexed by conjugacy classes of minimal sections of G, i.e. pairs
(T,S) of subgroups of G, where S is a normal subgroup of T contained in the
Frattini subgroup of T. These idempotents are orthogonal, and their sum is
equal to the identity. It follows that for any biset functor F over R, the
evaluation F (G) splits as a direct sum of specific R-modules indexed by
minimal sections of G, up to conjugation. The restriction of these
constructions to the biset category of p-groups, where p is a prime number
invertible in R, leads to a decomposition of the category of p-biset functors
over R as a direct product of categories F\_L indexed by atoric p-groups L up
to isomorphism. We next introduce the notions of L-enriched biset and
L-enriched biset functor for an arbitrary finite group L, and show that for an
atoric p-group L, the category F\_L is equivalent to the category of L-enriched
biset functors defined over elementary abelian p-groups. Finally, the notion of
vertex of an indecomposable p-biset functor is introduced (when p is invertible
in R), and, when R is a field of characteristic different from p, the objects
of the category F\_L are characterized in terms of vertices of their
composition factors.
