Moore-Penrose Dagger Categories

JS Lemay – 10 May 2023

The notion of a Moore-Penrose inverse (M-P inverse) was introduced by Moore in 1920 and rediscovered by Penrose in 1955. The M-P inverse of a complex matrix is a special type of inverse which is unique, always exists, and can be computed using singular value decomposition. In a series of papers in the 1980s, Puystjens and Robinson studied M-P inverses more abstractly in the context of dagger categories. Despite the fact that dagger categories are now a fundamental notion in many areas, the notion of a M-P inverse has not (to our knowledge) been revisited since their work. One purpose of this research project is to renew the study of M-P inverses in dagger categories.

In this talk, I will discuss the notion of a Moore-Penrose dagger category and provide many examples. I will also discuss generalized versions of singular value decomposition, compact singular value decomposition, and polar decomposition for maps in a dagger category, and show how, having such a decomposition is equivalent to having M-P inverses.

This is joint work with Robin Cockett, based on our QPL2023 proceedings paper: