III The Evolution of Possibility: 3 Category Theory as Formalisation of Potential

Having established that potentials are structured relational systems and actualisation is a perspectival shift, we now introduce a formal framework to model these dynamics: category theory. In relational ontology, category theory is not a mathematical abstraction applied to pre-existing objects; it is a tool for modelling structured potentials and their actualisations.

Categories as Systems of Potential

• A category is a network of objects (potentials) and morphisms (relations or constraints between them).

• This captures the structured nature of potentials, showing how one potential relates to another and how patterns of actualisation can propagate.

• Categories provide a map of intelligibility, not a depiction of temporal events or material substrates.

Functors as Constrained Perspectival Shifts

• A functor maps one category (system of potentials) to another, respecting the relational structure.

• Functors model perspectival shifts, showing how the same network of potentials can be actualised differently under distinct constraints.

• They formalise systematic transformations of relational intelligibility, linking different modes of actualisation.

Natural Transformations as Meta-Operations

• Natural transformations describe the relationships between functors themselves.

• They capture higher-order operations on construals, allowing us to compare and evolve systems of potentials.

• Through natural transformations, we can model co-individuation, evolution of patterns, and the emergence of regularities.

Implications for the Evolution of Possibility

• Category theory formalises the structure, actualisation, and transformation of potentials, providing a rigorous lens on relational evolution.

• It unites cosmology, biology, and semiotics under a single framework of intelligibility, showing how patterns, laws, and structures emerge naturally from relational potentials.

• With this formal apparatus, we are ready to explore differentiation and individuation, the next stage in the evolution of possibility.