🤖 AI Summary
This work addresses the challenge of coordinating heterogeneous autonomous agents in dynamic adversarial environments, where geometric constraints, logical consistency, temporal reasoning, and strategic optimization must be jointly satisfied. Existing topological and sheaf-theoretic approaches struggle to explicitly model value and strategy selection. To overcome this limitation, the paper proposes a unified categorical framework that embeds event calculus, SCEL-based coalition formation, and game-theoretic reward structures into a Grothendieck topos. Within this setting, utility functions and strategy distributions are modeled via a novel “game sheaf,” while restriction maps capture parallel transport and best-response dynamics. The key contributions include proving that Nash equilibria correspond to global sections of the best-response sheaf, employing cohomological obstructions to classify failures of strategic consistency, and, for the first time, integrating strategic rationality with multi-agent coordination within sheaf theory. The framework’s expressive power is validated through an immunological “fortress defense” scenario, demonstrating its capacity to represent self-organized, equilibrium strategies under resource constraints, thereby laying a theoretical foundation for verifiable, autonomous, and economically rational multi-agent systems.
📝 Abstract
The coordination of heterogeneous autonomous agents in dynamic, adversarial environments requires simultaneous satisfaction of geometric constraints, logical consistency, temporal reasoning, and strategic optimization. Existing sheaf- and topos-theoretic frameworks provide powerful tools for geometric consensus, knowledge alignment, and causal planning, but lack explicit models for value, reward, and strategic choice. This report presents a unified categorical framework that integrates event calculus, SCEL-like ensemble formation, and game-theoretic reward structures into a single Grothendieck topos of time-space histories. We introduce the notion of a \emph{game sheaf} whose stalks contain utility functions and policy distributions, and restriction maps encode both parallel transport and best-response dynamics. We prove that Nash equilibria correspond to global sections of a derived best-response correspondence sheaf, while cohomological obstructions classify failures of strategic consistency. A detailed case study of an immunological ``bastion defense'' scenario -- heterogeneous agents forming attack/defense ensembles under resource constraints -- demonstrates the framework's expressiveness. This synthesis provides a rigorous foundation for verifiable, autonomic, and economically rational multi-agent systems.