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Networks of Causal Abstractions: A Sheaf-theoretic Framework

A core challenge in causal artificial intelligence is the principled coordination of multiple, imperfect, and subjective causal perspectives arising from distributed agents with limited and heterogeneous access to the environment. This problem has received little formal treatment, as the existing framework assumes a single shared global causal model. This work introduces the causal abstraction network (CAN), a general sheaf-theoretic framework for representing, learning, and reasoning across collections of mixture of causal models (MCMs) - a class that unifies several existing models of context-dependent causal mechanisms. Sheaf theory provides a natural foundation for this task, offering a rigorous framework to coherently align distributed causal knowledge without requiring explicit causal graphs, functional mechanisms, interventional data, or jointly sampled observations. At the theoretical level, we provide a categorical formulation of MCMs and characterize key properties of CANs, including consistency and smoothness. Under consistency, we establish necessary and sufficient conditions: (i) for the existence of global sections, linked to spectral properties of an associated connection Laplacian; and (ii) for the convergence of causal knowledge diffusion over the CAN to the space of global sections. At the methodological level, we exploit the compositionality of causal abstractions to decompose the learning of consistent CANs into local problems on network edges, extending our prior work on Gaussian variables to Gaussian mixtures via the proposed MIXTURE-CALSEP algorithm. We validate the framework on synthetic data and through a financial application involving a multi-agent trading system, demonstrating CAN recovery, CAN-based portfolio optimization, and counterfactual reasoning.

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