  In "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, we
introduce the Frobenius P-categories giving two quite different definitions of
them. In this paper, we exhibit a third equivalent definition based on the form
of the old Alperin Fusion Theorem; this theorem can be reformulated in our
abstract setting, and ultimately depends on the behavior of the so-called
F-essential subgroups of P: we call "Alperin condition" a sufficient form of
this behavior. Then, we prove that a divisible P-category F is a Frobenius
P-category if and only if all the partial normalizers of a suitable set of
representatives for the F-isomorphism classes of subgroups of P fulfill both
the Sylow and the Alperin conditions.
