Non-unital $C^{*}$-categories, (co)limits, crossed products and exactness

TL;DR

This paper explores the foundational properties of non-unital C*-categories, extending crossed product constructions to categories with group actions, and demonstrates their preservation of exact sequences and equivalences.

Contribution

It provides a comprehensive reference for basic categorial properties of non-unital C*-categories and generalizes crossed product constructions to these categories.

Findings

Crossed product functor preserves exact sequences

Crossed product functor preserves excisive squares

Weak equivalences are sent to equivalences

Abstract

We provide a reference for basic categorial properties of the categories of (possibly non-unital) $\mathbb{C}$-linear $*$-categories or $C^{*}$-categories, and (not necessarily unit-preserving) functors. Generalizing the classical case of algebras with $G$-action, we extend the construction of crossed products to categories with $G$-action. We will show that the crossed product functor preserves exact sequences and excisive squares and sends weak equivalences to equivalences.