Overview
We model agents (individuals) in a relational field using category theory. Each agent is represented as an object, and morphisms denote emotional transmissions. Triggers introduce interference, while internal stability enables resonance. This is grounded in neurobehavioral principles: emotional regulation improves signal clarity, while dysregulation introduces reactive noise that destabilizes relational synchrony.
Category of Relational Agents
Objects: represent agents with emotional states and neurocognitive structures.
Morphisms: represent emotional transmissions (e.g., messages, behaviors, energetic influence).
Coherence Function: , where:
: perfect resonance and clarity.
: high interference and reactivity.
Internal Transformation
Let represent self-regulation or emotional stabilization. Self-regulation increases the coherence of outgoing morphisms.
Definition (Resource-Adaptive Mirror Agent, RAMA): Agent is a RAMA if for all morphisms , the coherence increases with internal regulation. That is, if stabilizes, outgoing signals become clearer.
Neurobehavioral Substrate
Emotional regulation via vagal tone and prefrontal activation reduces sympathetic reactivity and increases signal predictability.
High-coherence morphisms correlate with synchronized neural oscillations and improved inter-brain coupling.
Dysregulated agents produce erratic, high-frequency noise, disrupting neural entrainment.
Adaptive agents reduce fear-based responses, allowing safe mirroring and mutual tuning.
Relational Field
Let be a presheaf encoding potential states of communication between agents and . When both agents stabilize, morphisms between them become naturally isomorphic.
Emotional State Functor
Define a functor , where is a category of vibrational/emotional frequencies.
Resonance
A natural transformation exists when agents emotionally tune. This requires:
Internal quieting of (e.g., limbic-prefrontal recalibration),
Removal of emotional interference (e.g., trauma, triggers),
Stability across morphisms (trust signals, coherence).
Relational Morphisms Between RAMAs
Relational dynamics are captured as functors between the sheaves modeling each agent: This functor encodes the influence of RAMA ’s state on RAMA . Bidirectional tuning is modeled via an adjunction: Mutual regulation emerges when these functors approach equivalence.
Bidirectional Functor Pairs and Dyadic RAMAs
When both agents co-adapt, the bidirectional functor pair becomes a natural isomorphism: This corresponds to high attunement and shared regulatory rhythm, forming a relational dyad within the RAMA topology.
Homotopy and Operadic Synchronization
Synchrony emerges not only in isolated dyads but in larger groupings. Using operads , we define emotional synchronization over multiple agents. Homotopy between functors models gradual realignment: where is a group field state.
Internal Logic and Trauma Obstruction
Each RAMA sheaf carries a local internal logic , modeled by an internal language of a topos. Trauma or emotional rigidity appears as an obstruction in gluing—non-extendable local sections or broken sheaf conditions. These are formalized as monomorphisms with no section.
Relational Topologies
The site topology may be refined based on relational coherence. We define a new Grothendieck topology where covers correspond to trustful, bidirectional morphism clusters. Agents with high mutual coherence form the open sets.
Dyadic Comonads and Adaptive Pressure Loops
Adaptive loops are modeled with comonads , where:
is the comonadic context (shared field pressure),
models recursive feedback (e.g., mirroring back responses),
models momentary extraction (e.g., attentional attunement).
Dyadic RAMAs share a comonadic context when interacting closely.
Conflict, Repair, and Resilience Morphisms
Conflicts appear as disruptions in coherence: . Repair involves the reintroduction of stabilizing morphisms such that . Resilience is modeled by the ability to restore natural transformations after perturbation:
Unified Diagram (Placeholder)
Conclusion
If through self-regulation, then the coherence of increases. If also adapts, a natural isomorphism emerges: This represents emotional resonance across a relational field. The RAMA framework demonstrates how shifts in one agent’s internal field can ripple through to others, enabling co-regulation, attunement, and the emergence of shared coherence.