Moore-Penrose Dagger Categories
Robin Cockett (University of Calgary), Jean-Simon Pacaud Lemay, (Macquarie University)

TL;DR
This paper extends the concept of Moore-Penrose inverses to dagger categories, introducing Moore-Penrose dagger categories and characterizing maps with M-P inverses using generalized decompositions.
Contribution
It introduces Moore-Penrose dagger categories, provides multiple examples, and establishes equivalences between M-P inverses and generalized decompositions in dagger categories.
Findings
Defined Moore-Penrose dagger categories.
Established equivalence between M-P inverses and generalized decompositions.
Characterized maps with M-P inverses in various dagger categories.
Abstract
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 categorical quantum mechanics, the notion of a M-P inverse has not (to our knowledge) been revisited since their work. One purpose of this paper is, thus, to renew the study of M-P inverses in dagger categories. Here we introduce the notion of a Moore-Penrose dagger category and provide many examples including complex matrices, finite Hilbert spaces, dagger groupoids, and inverse categories. We also introduce…
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