Abstract
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.
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