Categorical Quantum Mechanics - constructing the category of homogenous cones

Abstract

Category theory provides a unifying language to reason about diverse structures in mathematics and physics. Its foundational perspective proves especially potent for describing and generalizing frameworks in physical systems, such as those mod- eled by Generalized Probabilistic Theories (GPTs). This thesis bridges categorical thinking with the study of homogeneous cones, which serve as a central mathemat- ical representation for state spaces in GPTs. Specifically, we establish a categorical equivalence between the category of homogeneous cones equipped with an order unit and the category of T-algebras. This equivalence highlights the power of cate- gory theory to not only organize known physical structures but also to suggest novel pathways for reasoning about physical phenomena and theoretical generalizations. These results reinforce the role of categorical methods as a natural and powerful toolkit for modern theoretical physics