  In the first part of this paper, we give a new look at inclusions of von
Neumann algebras with finite-dimensional centers and finite Jones' index. The
minimal conditional expectation is characterized by means of a canonical state
on the relative commutant, that we call the spherical state; the minimal index
is neither additive nor multiplicative (it is submultiplicative), contrary to
the subfactor case. So we introduce a matrix dimension with the good functorial
properties: it is always additive and multiplicative. The minimal index turns
out to be the square of the norm of the matrix dimension, as was known in the
multi-matrix inclusion case. In the second part, we show how our results are
valid in a purely 2-$C^*$-categorical context, in particular they can be
formulated in the framework of Connes' bimodules over von Neumann algebras.
