Morita homotopy theory of C*-categories

TL;DR

This paper develops a homotopy theory framework for small unital C*-categories, establishing a model structure, describing the Picard group, and exploring the additive properties of the associated homotopy category.

Contribution

It constructs a cofibrantly generated model structure on C*-categories, characterizes the Morita homotopy category, and links it to classical invariants like the Picard group and Grothendieck group.

Findings

Established a model structure with Morita equivalences as weak equivalences.

Described the Picard group within the homotopy category.

Proved the homotopy category is semi-additive and related to the Grothendieck group.

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.