A dagger kernel category of complete orthomodular lattices
Michal Botur, Jan Paseka, Richard Smolka
TL;DR
The paper develops ${\bf SupOMLatLin}$, the category of complete orthomodular lattices with linear maps, as a dagger kernel category with dagger biproducts and free objects, drawing strong parallels to quantum-logical structures. It proves that every morphism $f:X\to Y$ admits an essentially unique zero-epi followed by a dagger monomorphism factorization through the principal downset ${\mathop{\downarrow}}f(1)$, and it analyzes kernels, cokernels, and Sasaki projections to anchor this factorization. The work provides explicit constructions of dagger kernels, the zero-epi–kernel factorization, and arbitrary dagger biproducts, while showing that free objects are given by complete powerset algebras $\mathcal{P}(A)$ with a universal property. It also points toward a quantaloid interpretation by linking endomorphisms to a potential quantale ${\bf Lin}(X)$, signaling deeper connections between complete orthomodular lattices and categorical quantum logic.
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.