On Krull-Schmidt bicategories

TL;DR

This paper introduces Krull-Schmidt bicategories, establishing conditions for unique decompositions into indecomposables and linking these to idempotent splitting and endomorphism properties, with applications in 2D linear representation theory.

Contribution

It provides a simple definition of Krull-Schmidt bicategories and proves the uniqueness of decompositions, extending classical results to bicategorical contexts.

Findings

Unique decomposition into indecomposables is guaranteed in Krull-Schmidt bicategories.

Characterization of these bicategories via idempotent splitting and endomorphism rings.

Numerous examples from 2-dimensional linear representation theory.

Abstract

We study the existence and uniqueness of direct sum decompositions in additive bicategories. We find a simple definition of Krull-Schmidt bicategories, for which we prove the uniqueness of decompositions into indecomposable objects as well as a characterization in terms of splitting of idempotents and properties of 2-cell endomorphism rings. Examples of Krull-Schmidt bicategories abound, with many arising from the various flavors of 2-dimensional linear representation theory.