A Dagger Kernel Category of Complete Orthomodular Lattices

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 \({\textbf {SupOMLatLin}}\) of complete orthomodular lattices with linear maps. We demonstrate that \({\textbf {SupOMLatLin}}\) itself forms a dagger kernel category, equipped with additional structure such as dagger biproducts and free objects. A key result establishes a concrete description of how every morphism in \({\textbf {SupOMLatLin}}\) 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 \({\textbf {SupOMLatLin}}\), provides new insights into the interplay between complete orthomodular lattices and the foundational concepts of quantum theory.