A structural approach in cognitive developmental research with category theory

This paper presents a novel mathematical framework for cognitive development theory using category theory. First, we introduce formalization of Piaget's concept of groupings. We address mathematical inconsistencies in Piaget's structuralism by reconceptualizing groupings as specific groupoids where objects form partial join-semilattices, with reversibility represented as morphism invertibility and compositional operations as lattice operations. Then, we propose a new line of research program about children’s subjective experience using category theory. Specifically, we propose a line of empirical research comparing children's and adults' subjective experiences through a structure analysis. Our first step of this program has already revealed a counterintuitive discrepancy between children’s color naming abilities and color similarity structures, suggesting fundamental importance in distinguishing subjective experience and reports about it in cognitive developmental research. This category theory based structural approach opens possibilities for unified theories as interconnected categorical domains, potentially enabling prediction of developmental trajectories and identification of critical intervention periods through precise mathematical relationships governing cognitive architecture.