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arXiv · 1112.5563

Morita homotopy theory of C*-categories

Ivo Dell'Ambrogio·Goncalo Tabuada
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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 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.

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Ivo Dell'Ambrogio, Goncalo Tabuada. 2011-12-23. Morita homotopy theory of C*-categories. https://doi.org/10.1016/j.jalgebra.2013.09.022

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