AI summaryⓘ
The authors present a new mathematical framework to help different autonomous agents work together in complex and changing environments where they face challenges from opponents. They combine ideas from geometry, logic, time, and game theory into one structure called a 'game sheaf,' which helps model how agents make decisions and respond to each other strategically. They show that finding stable strategies (Nash equilibria) fits naturally into this framework, and they use a detailed example related to immune system defense to illustrate their approach. This work aims to create a solid foundation for building multi-agent systems that can act logically, cooperatively, and in their own best interests.
Grothendieck topossheaf theorygame theoryNash equilibriumevent calculusautonomous agentspolicy distributioncohomologystrategic optimizationensemble formation
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.