A dagger kernel category of complete orthomodular lattices

TL;DR

This paper shows that the category of complete orthomodular lattices with linear maps forms a dagger kernel category, revealing new structural insights relevant to quantum logic and foundational quantum theory.

Contribution

It establishes that SupOMLatLin is a dagger kernel category with unique morphism factorizations, advancing the mathematical framework for quantum logic within category theory.

Findings

SupOMLatLin is a dagger kernel category.

Every morphism admits a unique zero-epi and dagger monomorphism factorization.

The structure enhances understanding of quantum logic and foundational quantum theory.

Abstract

Dagger kernel categories, a powerful framework for studying quantum phenomena within category theory, provide a rich mathematical structure that naturally encodes key aspects of quantum logic. This paper focuses on the category SupOMLatLin of complete orthomodular lattices with linear maps. We demonstrate that SupOMLatLin itself forms a dagger kernel category, equipped with additional structure such as dagger biproducts and free objects. A key result establishes that every morphism in SupOMLatLin admits an essentially unique factorization as a zero-epi followed by a dagger monomorphism. This factorization theorem, along with the dagger kernel category structure of SupOMLatLin, provides new insights into the interplay between complete orthomodular lattices and the foundational concepts of quantum theory.