Morita homotopy theory of C⁎-categories

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 (denoted by MMor) on the category C1⁎cat 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 Brown-Green-Rieffelʼs Picard group in the associated homotopy category Ho(MMor). We then prove that Ho(MMor) is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category Ho(MMor)−1 and a composite functor C1⁎cat→Ho(MMor)→Ho(MMor)−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(MMor)−1 by the tensor unit object.