A Meta-Categorical Theory of Causal Emergence and Hierarchical Information Complexity in Distributed Systems: An Introductory Axiomatization

We present a preliminary mathematical framework for formalizing causal emergence and hierarchical information complexity in distributed systems using categorical methods. This introductory work develops an axiomatic foundation that integrates category theory, information theory, and homological algebra to address the formal characterization of multi-scale causality. While comprehensive, this paper represents only initial steps toward a complete theory, with significant theoretical and computational challenges remaining. We establish five fundamental axioms, prove basic existence results for finite Markov systems, and demonstrate computational feasibility through controlled examples. Our approach aims to provide rigorous mathematical tools for understanding how causal relationships transform across hierarchical scales, though much work remains to fully realize this ambitious program. MSC Classifications: 18A05, 18A15, 94A17, 68Q87, 55N35