Category Theory Through the Lens of Relational Ontology: 2 Morphisms as Actualisation Pathways

Category theory tells us that morphisms are arrows between objects.

But once objects are reconceived as cuts in potential, the nature of a morphism changes entirely.

A morphism is not a map.

A morphism is not a function.

A morphism is not a piece of syntax connecting two fixed points.

A morphism is a movement: the shift in horizon configuration that carries one stabilised cut into another.

To treat morphisms this way is not an interpretive flourish.

It is a redefinition that follows necessarily from taking relation as ontologically primitive.

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1. The False Comfort of “Maps”

In conventional presentations, morphisms are introduced by analogy:

• “a function from set A to set B,”

• “a homomorphism preserving structure,”

• “a mapping doing something to objects.”

These heuristics assume that objects pre-exist and morphisms act upon them.

But this is an artefact of substance metaphysics.

If objects are relationally stabilised cuts, then a morphism cannot “go between” them.

It must transform one relational configuration into another.

A morphism is a reconfiguration, not a correspondence.

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2. Morphisms as Horizon Movements

Let X and Y be two cuts: stabilised regions of potential within a wider relational field.

Then a morphism

  f : X → Y

is the actualisation pathway that:

• shifts the horizon from the stance encoded in X

• toward the stance stabilised as Y,

• without collapsing the viability of the relational field during the transition.

We can phrase this operationally:

A morphism is an energetically permitted shift in readiness from one perspectival contraction to another.

This makes morphisms dynamic, not static.

They are trajectories in the field of possible construals.

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3. Morphisms Have Cost

Once we treat morphisms as actualisation pathways, we uncover something absent from classical category theory:

every morphism has metabolic cost.

There is no free shift in readiness.

To move from one horizon configuration to another:

• gradients must be preserved,

• incompatibilities must be resolved,

• the system must spend energy maintaining stability during the transition.

This aligns category theory with our model of mind, biology, and cosmology:

actualisation is always constrained and always costly.

Even identity morphisms (as shown in Post 1) are maintenance work.

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4. Composition as Sequential Actualisation

If

  f : X → Y

  g : Y → Z

are actualisation pathways, then the composition

  g ∘ f : X → Z

is the viable concatenation of two horizon shifts.

Composition is not “function composition.”

It is the continuity condition for multi-step perspectival movement:

• the end state of f must be metabolically and relationally compatible with the starting conditions of g;

• the shift from Y to Z must not contradict what was stabilised when moving from X to Y.

This immediately makes the associativity law lucid:

Perspectival transitions must form stable sequences across multiple cuts.

Associativity is not an axiom.

It is a constraint imposed by the ontology.

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5. Morphisms Encode Affordance, Not Content

Because morphisms are horizon movements, they encode:

• what can follow from a given cut,

• what transformations of readiness are available,

• what potential is supported by the ecological configuration,

• what actualisation pathways are viable.

They do not encode:

• information,

• semantic content,

• representational tokens,

• computational operations.

A morphism is an affordance, not a message.

This is precisely where category theory aligns with biological evolution and the Cognitive Thread:

organisms succeed not by representing the world

but by tracking viable transitions between states of readiness.

Morphisms are that logic.

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6. Inverse Morphisms and Reversibility

Standard category theory treats invertible morphisms as isomorphisms — structural equivalences.

In a relational ontology:

invertibility becomes reversibility of horizon configuration, which is extremely rare.

Most actualisation pathways are not reversible:

• metabolic gradients dissipate,

• stabilised cuts lose fidelity,

• construal events change the potential field.

Thus:

• isomorphisms represent high-stability, low-cost, reversible cuts

• non-isomorphisms represent irreversible or costly perspectival transitions

• epimorphisms/monomorphisms recut as different kinds of horizon constraint (to be formalised in later posts)

This provides a natural ecological interpretation of categorical structure.

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7. Why This Matters

Once morphisms are re-understood as actualisation pathways:

• category theory becomes a theory of viable transformations,

• composition becomes horizon ecology,

• identity becomes resilience,

• invertibility becomes energetic symmetry,

• the entire framework aligns with our cosmology, biology, and cognition.

This recut allows us to treat category theory not as a formal language about structures

but as a formal language for the dynamics of relation itself.

It becomes the mathematics of what moves, what can move, and what cannot.

And this prepares the ground for the next post.

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Next: Post 3 — Functors as Cross-Scale Translations of Relational Horizon