TY - GEN
T1 - Moore-Penrose dagger categories
AU - Cockett, Robin
AU - Lemay, Jean-Simon Pacaud
N1 - © R. Cockett & J.-S. P. Lemay. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.
PY - 2023/8/30
Y1 - 2023/8/30
N2 - 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 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 allows us to provide precise characterizations of which maps have M-P inverses in a dagger idempotent complete category, a dagger kernel category with dagger biproducts (and negatives), and a dagger category with unique square roots.
AB - 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 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 allows us to provide precise characterizations of which maps have M-P inverses in a dagger idempotent complete category, a dagger kernel category with dagger biproducts (and negatives), and a dagger category with unique square roots.
UR - http://www.scopus.com/inward/record.url?scp=85173501596&partnerID=8YFLogxK
UR - https://cgi.cse.unsw.edu.au/~eptcs/paper.cgi?QPL2023.10
U2 - 10.4204/EPTCS.384.10
DO - 10.4204/EPTCS.384.10
M3 - Conference proceeding contribution
AN - SCOPUS:85173501596
T3 - Electronic Proceedings in Theoretical Computer Science, EPTCS
SP - 171
EP - 186
BT - Proceedings of the Twentieth International Conference on Quantum Physics and Logic
A2 - Mansfield, Shane
A2 - Valiro, Benoît
A2 - Zamdzhiev, Vladimir
PB - Open Publishing Association
CY - Waterloo, NSW
T2 - 20th International Conference on Quantum Physics and Logic, QPL 2023
Y2 - 17 July 2023 through 21 July 2023
ER -