Moore-Penrose Dagger Categories

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.Comment: In Proceedings QPL 2023, arXiv:2308.1548

On difunctions

The notion of a difunction was introduced by Jacques Riguet in 1948. Since then it has played a prominent role in database theory, type theory, program specification and process theory. The theory of difunctions is, however, less known in computing than it perhaps should be. The main purpose of the current paper is to give an account of difunction theory in relation algebra, with the aim of making the topic more mainstream. As is common with many important concepts, there are several different but equivalent characterisations of difunctionality, each with its own strength and practical significance. This paper compares different proofs of the equivalence of the characterisations. A well-known property is that a difunction is a set of completely disjoint rectangles. This property suggests the introduction of the (general) notion of the “core” of a relation; we use this notion to give a novel and, we believe, illuminating characterisation of difunctionality as a bijection between the classes of certain partial equivalence relations

Coalgebraische Similarität

Bereits bekannt erhält jeder Funktor genau dann schwache Pullbacks, wenn jede Kongruenz eine difunktionale Bisimulation ist. In Kapitel 3 fanden wir äquivalente Aussagen für die schwache Kerpaarerhaltung und die Urbilderhaltung. Ausserdem definierten wir eine Funktorabänderung, die wir Urbildbereinigung nannten. Der resultierende Funktor erhält Urbilder. Die Idee war inspiriert von der Transformation, so dass daraus ein gesunder Funktor entsteht. Der Urbilder erhaltende Funktor hat auch den Vorteil, dass seine Unterfunktoren genau die Urbilder erhaltende Unterfunktoren des ursprünglichen Funktors sind. In Kapitel 4 zeigten wir, dass die monotonen trennbaren Boxen eine korrekte und vollständige Modallogik liefern. Interessant ist, dass die Urbild-Bereinigung des allgemeinen Nachbarschaftsfunktors einen Funktor liefert, der schwache Pullbacks erhält