If dialogue is recursive actualisation within structured potential, and if cuts restructure that potential, and if asymmetry sustains generativity, then we require a formal language capable of modelling:
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positions without reifying them as substances,
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transformation without invoking hidden essences,
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structure without collapsing into static representation.
Category theory offers such a language — not because it describes things, but because it privileges relations and transformations.
This movement is not a technical exercise. It is a structural sketch.
1. From Substances to Positions
In classical metaphysics, individuals are primary and relations secondary.
In relational ontology, this priority is inverted. Positions emerge within a field of structured potential.
Category theory begins from a similar inversion:
Objects are not defined by intrinsic content but by their relations — by the morphisms they participate in.
We may therefore model a dialogical situation as follows:
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A position within dialogue corresponds to an object.
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A cut corresponds to a morphism.
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The structured potential corresponds to the category itself.
An object is nothing over and above its place in a web of possible transformations.
So too, a dialogical position is nothing over and above its gradients of constraint and possibility within the relational field.
2. Cuts as Morphisms
Each utterance — each perspectival cut — can be modelled as a morphism:
This does not mean that a fixed content travels from A to B.
Rather:
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A is a position prior to the cut.
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B is the reconfigured position after the cut.
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f is the transformation enacted by the utterance.
The morphism does not transport substance. It alters relational structure.
Crucially, morphisms compose.
If:
then:
Dialogue is therefore not a chain of isolated events. It is compositional structure. Each cut builds upon previous cuts, forming trajectories through the field.
History is composition.
3. The Category as Structured Potential
The category itself represents the structured potential of the dialogue:
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The objects are possible positions.
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The morphisms are possible cuts.
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Composition encodes recursive actualisation.
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Identity morphisms represent stabilisation of a position without transformation.
But note carefully:
The category is not static background. It is continuously restructured through the addition of new morphisms.
Strong construal events alter the topology of the category itself. They introduce new pathways, collapse old ones, or reconfigure equivalences.
Thus the system evolves through its own instances.
The category is historical.
4. Asymmetry in Categorical Terms
Dialogue is not symmetric exchange.
In categorical language:
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Morphisms need not be invertible.
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Not every cut can be undone.
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Composition is directional.
Irreversibility reflects the historical accumulation of cuts. Once a transformation has occurred, the field cannot return to its prior undifferentiated state.
Asymmetry is therefore not anomaly. It is structural necessity.
If every morphism were invertible, dialogue would reduce to trivial equivalence. Nothing would genuinely differentiate.
Non-invertibility preserves ontological weight.
5. Perturbation as Non-Commutativity
In many categories, composition is non-commutative:
Order matters.
So too in dialogue.
A reframing introduced early may produce a radically different trajectory than the same reframing introduced later.
Perturbations alter the path-dependence of the system.
Non-commutativity models this sensitivity.
Dialogue is not merely what is said, but when and how it is cut into the field.
6. Non-Fusion and Distinct Objects
Co-individuation does not collapse objects into one.
Even if two positions become highly aligned, they remain distinct objects within the category — connected by rich morphisms, but not identical.
Fusion would correspond to categorical identification — the collapse of distinct objects into one via isomorphism.
But genuine dialogue resists total isomorphism.
If two positions were perfectly isomorphic, no further morphisms would generate new structure.
Individuation would stall.
Difference must persist for transformation to remain meaningful.
7. Functorial Drift: Dialogue Across Contexts
Dialogue does not occur in isolation. It maps onto broader systems — cultural, linguistic, political.
We may model this through functors:
A functor maps one category to another, preserving structure.
A local dialogue (Category D) may map into a broader discursive system (Category C).
Cuts within D induce structural shifts within C — but not always identically. The mapping may preserve composition but reinterpret objects.
This models how private dialogue can reshape public discourse, or how institutional structures constrain individual exchange.
Dialogue is nested within larger relational architectures.
8. The Limits of Formalisation
This sketch is not meant to mathematise conversation.
It serves one purpose:
To demonstrate that dialogue can be rigorously conceived without reverting to substance metaphysics.
Category theory shows that:
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transformation can be primary,
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relation can define identity,
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structure can emerge through composition,
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and asymmetry can be formal rather than accidental.
9. Dialogue as Structured Becoming
Across these four movements, we have traced:
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Co-individuation as recursive actualisation.
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The cut as perspectival transformation.
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Asymmetry and perturbation as conditions of generativity.
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Morphic composition as formal architecture.
Dialogue emerges not as communication between substances but as structured becoming within relational potential.
Ontology is not elsewhere.
It is enacted here.
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