  In this article we establish the foundations of the Morita homotopy theory of
C*-categories. Concretely, we construct a cofibrantly generated simplicial
symmetric monoidal Quillen model structure M_Mor on the category C*cat1 of
small unital C*-categories. The weak equivalences are the Morita equivalences
and the cofibrations are the *-functors which are injective on objects. As an
application, we obtain an elegant description of the Brown-Green-Rieffel Picard
group in the associated Morita homotopy category Ho(M_Mor). We then prove that
the Morita homotopy category is semi-additive. By group completing the induced
abelian monoid structure at each Hom-set we obtain an additive category
Ho(M_Mor)^{-1} and a canonical functor C*cat1 {\to} Ho(M_Mor)^{-1} which is
characterized by two simple properties: inversion of Morita equivalences and
preservation of all finite products. Finally, we prove that the classical
Grothendieck group functor becomes co-represented in Ho(M_Mor)^{-1} by the
tensor unit object.
