Morita homotopy theory of C*-categories

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 M[subscript Mor]) 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(M[subscript Mor]). We then prove that Ho(M[subscript Mor]) is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category Ho(M[subscript Mor])[superscript −1] and a composite functor C1*cat→Ho(M[subscript Mor][superscript −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[subscript Mor])[superscript −1] by the tensor unit object.