Shared Protentions in Multi-Agent Active Inference

In this paper, we unite concepts from Husserlian phenomenology, the active inference
framework in theoretical biology, and category theory in mathematics to develop a comprehensive
framework for understanding social action premised on shared goals. We begin with an overview
of Husserlian phenomenology, focusing on aspects of inner time-consciousness, namely, retention,
primal impression, and protention. We then review active inference as a formal approach to modeling
agent behavior based on variational (approximate Bayesian) inference. Expanding upon Husserl’s
model of time consciousness, we consider collective goal-directed behavior, emphasizing shared
protentions among agents and their connection to the shared generative models of active inference.
This integrated framework aims to formalize shared goals in terms of shared protentions, and thereby
shed light on the emergence of group intentionality. Building on this foundation, we incorporate
mathematical tools from category theory, in particular, sheaf and topos theory, to furnish a mathe-
matical image of individual and group interactions within a stochastic environment. Specifically, we
employ morphisms between polynomial representations of individual agent models, allowing pre-
dictions not only of their own behaviors but also those of other agents and environmental responses.
Sheaf and topos theory facilitates the construction of coherent agent worldviews and provides a way
of representing consensus or shared understanding. We explore the emergence of shared protentions,
bridging the phenomenology of temporal structure, multi-agent active inference systems, and cate-
gory theory. Shared protentions are highlighted as pivotal for coordination and achieving common
objectives. We conclude by acknowledging the intricacies stemming from stochastic systems and
uncertainties in realizing shared goals.

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