[Paper Review] Colimits in the correspondence bicategory

Suliman Albandik, Ralf Meyer|arXiv (Cornell University)|Feb 26, 2015

Advanced Operator Algebra Research17 references6 citations

TL;DR

This paper establishes that key C*-algebra constructions—such as crossed products, Cuntz–Pimsner algebras, inductive limits, and amalgamated free products—can be universally characterized as colimits in the bicategory of C*-correspondences. By interpreting these constructions via universal properties in a bicategorical framework, the authors unify their categorical structure and show that proper correspondences yield colimits via natural equivalences of groupoids, generalizing classical C*-algebra limits and providing a conceptual foundation for noncommutative quotients.

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.

Motivation & Objective

• To provide a unified bicategorical framework for understanding fundamental C*-algebra constructions using colimits.
• To demonstrate that constructions like crossed products and Cuntz–Pimsner algebras arise naturally as colimits in the bicategory Corr of C*-correspondences.
• To show that inductive limits of C*-algebras with *-homomorphisms are isomorphic to colimits in Corrprop when the system consists of proper correspondences.
• To establish that the universal property of colimits in Corr matches the standard definitions of known C*-algebraic objects, thereby proving their existence and uniqueness up to equivalence.

Proposed method

• The authors use the bicategory Corr of C*-correspondences, where objects are C*-algebras, arrows are Hilbert modules with compatible actions, and 2-arrows are unitary intertwiners.
• They define diagrams in Corr as functors from a small category C to the subbicategory Corrprop of proper correspondences.
• Colimits are characterized via a universal property: for any C*-algebra D, the category of transformations from a diagram F to the constant diagram on D is naturally equivalent to the groupoid of correspondences from the colimit to D.
• The construction relies on the fact that proper correspondences allow the formation of tensor products and that the universal property of colimits matches known C*-algebra constructions.
• The authors simplify diagrams indexed by N (inductive systems) to sequences of correspondences En+1_n, showing that such systems are equivalent to full diagrams up to isomorphism.
• They prove that inductive limits in the classical category of C*-algebras with *-homomorphisms coincide with colimits in Corrprop by constructing an equivalence of groupoids between correspondences into the limit and transformations from the diagram.

Experimental results

Research questions

• RQ1 Can standard C*-algebra constructions such as crossed products and Cuntz–Pimsner algebras be interpreted as colimits in a bicategorical framework?
• RQ2 How does the universal property of colimits in the bicategory Corr relate to the standard definitions of C*-algebraic constructions?
• RQ3 Under what conditions do inductive limits of C*-algebras with *-homomorphisms arise as colimits in Corrprop?
• RQ4 Is the Cuntz–Pimsner algebra of a proper product system isomorphic to the colimit of the corresponding diagram in Corr?
• RQ5 What is the role of proper correspondences in ensuring the existence of colimits in Corr?

Key findings

• The crossed product for a twisted group action arises as a colimit in the bicategory Corr, characterized by the same universal property as the standard crossed product.
• The Cuntz–Pimsner algebra of a proper product system is isomorphic to the colimit of the corresponding diagram in Corr, without passing through the Cuntz–Toeplitz algebra.
• Inductive limits of C*-algebras with *-homomorphisms are isomorphic to colimits in Corrprop, and the universal property of the inductive limit in the classical category matches the bicategorical colimit.
• The colimit of a diagram of proper correspondences indexed by a group is isomorphic to the full C*-algebra of sections of the associated Fell bundle.
• For inductive systems, the colimit in Corr is equivalent to the classical inductive limit in the category of C*-algebras, provided the system consists of proper correspondences.
• The construction of the colimit is functorial and respects isomorphisms, with the groupoid of correspondences into the colimit naturally equivalent to the groupoid of transformations from the diagram to any target C*-algebra.