Additive C*-categories and K-theory

We review the notions of a multiplier category and the $W^{}$-envelope of a $C^{}$-category. We then consider the notion of an orthogonal sum of a (possibly infinite) family of objects in a $C^{}$-category. Furthermore, we construct reduced crossed products of $C^{}$-categories with groups. We axiomatize the basic properties of the $K$-theory for $C^{}$-categories in the notion of a homological functor. We then study various rigidity properties of homological functors in general, and special additional features of the $K$-theory of $C^{}$-categories. As an application we construct and study interesting functors on the orbit category of a group from $C^{*}$-categorical data.

💡 Research Summary

This paper provides a foundational reference on three central themes in the theory of C*-categories: orthogonal sums of infinite families of objects, reduced crossed products with groups, and the axiomatization and rigidity properties of K-theory.

The first part rigorously develops the theory of orthogonal sums in C*-categories. Inspired by the classical direct sum of Hilbert C*-modules, the authors define the orthogonal sum of a (possibly infinite) family of objects in a unital C*-category via a universal property (Definition 5.15). This notion is characterized by describing spaces of bounded adjointable operators into and out of the sum. The paper shows that this abstract definition coincides with the classical construction in the category of Hilbert modules (Theorem 8.4) and is equivalent to a prior definition by Flath and Wurst. An alternative construction, the AV-sum (Definition 7.1) due to Antoun and Voigt, is also discussed and shown to behave better with multiplier categories. Techniques like a Yoneda-type embedding (Theorem 10.1) allow the embedding of any C*-category into one admitting all small sums.

The second major contribution is the construction of the reduced crossed product C ⋊_r G for a C*-category C with an action of a group G (Definition 12.9). This is achieved by completing the algebraic crossed product with respect to a norm derived from a representation on a C*-category L^2(G, C), whose morphism spaces themselves are defined using orthogonal sums indexed by G. Theorem 12.1 establishes the existence and basic properties of this reduced crossed product functor.

The third core theme is the study of homological functors, which axiomatize key properties of the K-theory functor for C*-categories (Definition 13.4). These axioms include additivity, behavior with respect to orthogonal sums, and annihilation of flasque categories. The authors verify that the K-theory functor K_{C*Cat} satisfies these axioms. Within this framework, they prove special rigidity properties of K-theory, most notably that it preserves arbitrary products of additive C*-categories (Theorem 15.7). Furthermore, the paper investigates how K_{C*Cat} behaves with respect to various equivalence notions: it preserves Morita equivalences (Theorem 16.18), sends relative Morita equivalences and relative idempotent completions to equivalences (Propositions 17.3, 17.7), sends Murray-von Neumann equivalent morphisms to equivalent morphisms (Proposition 17.13), and sends weak Morita equivalences (a functional-analytic notion defined in Definition 18.3) to equivalences (Theorem 18.6).

Finally, the paper synthesizes these tools in a significant application: the construction of equivariant homology theories on the orbit category G Orb of a group from C*-categorical data (Section 19). Given a unital C*-category C with strict G-action and an auxiliary functor (like K_{C*Cat}), two functors Hg^G_{C, max} and Hg^G_{C, r} are constructed. The first is defined homotopy-theoretically using maximal crossed products, while the second is given an explicit description using orthogonal sums and reduced crossed products. A comparison map between them is provided, and when C is the category of finitely generated projective Hilbert A-modules, the construction recovers the Davis-Lück functor (Proposition 19.18). This machinery is foundational for the study of the Baum-Connes assembly map in subsequent work.