Colimits in the correspondence bicategory

TL;DR

This paper explores how various constructions in C*-algebra theory, such as crossed products and Cuntz-Pimsner algebras, can be understood as colimits within the bicategory of correspondences, unifying different concepts.

Contribution

It provides a new categorical perspective by interpreting key C*-algebra constructions as colimits in the correspondence bicategory, offering a unifying framework.

Findings

Crossed products are characterized as colimits in the bicategory.

Cuntz-Pimsner algebras are interpreted as colimits of proper product systems.

Various algebraic constructions like direct sums and amalgamated free products are also realized as colimits.

Abstract

We interpret several constructions with C*-algebras as colimits in the bicategory of correspondences. This includes crossed products for actions of groups and crossed modules, Cuntz-Pimsner algebras of proper product systems, direct sums and inductive limits, and certain amalgamated free products.