Category Theory Through the Lens of Relational Ontology: 1 When Category Theory Begins With Relation, Not Objects

Category theory is often described as “the mathematics of structure,” or “the study of relationships rather than things.”

But even these slogans smuggle in the very metaphysics they claim to undo.

Standard category theory still begins with objects.

It treats them as given, as atomic units to be linked by morphisms.

Relation is secondary: a mapping between pre-posited entities.

What happens when we invert that order?

What happens when relation, not object, is primary?

What happens when objects are not “points,” but cuts in a field of potential?

This post is the beginning of that inversion.

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1. Objects Are Cuts: The First Relational Shift

In category theory, an object is usually treated as inert — a node with no internal structure, distinguished only by the morphisms it participates in.

In relational ontology, this is backward.

An “object” is not a unit.

It is a perspectival contraction of potential —

a stabilised cut in a relational field.

To be an object is to be a locally coherent reduction of possibility.

An object is not a thing.

It is a way the horizon holds still long enough for something to be done.

Thus, the category does not begin with objects.

It begins with potential, then cuts, and only then stable nodes of construal.

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2. Morphisms Are Actualisation Pathways

If objects are cuts in potential, morphisms cannot be “maps between things.”

A morphism is:

• a pathway of actualisation,

• a transformation in the horizon of readiness,

• the relational move that takes one stabilised cut to another.

Morphisms track changes in perspectival configuration, not information passed between entities.

Composition then becomes:

the compatibility of successive actualisation pathways.

A horizon can shift → and shift again → only if the second shift remains viable from the first.

This is category theory as metabolic ecology, not algebraic plumbing.

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3. Identity Morphisms as Persistence Conditions

Identity morphisms are usually treated as trivial: the arrow that does nothing.

But in a relational ontology, “doing nothing” is the most nontrivial fact of all.

An identity morphism is:

• the stability condition under which a cut persists across time,

• the maintenance of a horizon configuration against drift,

• the metabolic cost of keeping a potential open.

Identity is energetic, not formal.

It is a readiness-maintenance process.

Thus, id_X is not “the map from X to itself.”

It is the condition that X remains X in the face of perturbation.

This alone is enough to rewrite half of category theory.

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4. Composition as Horizon Coherence

Composition in standard category theory is purely formal.

But if morphisms are shifts in readiness, then composition is:

the coherence of horizon transformations across cuts.

A → B → C is viable only if:

• the contraction from A to B
  does not remove the gradient needed to reach C, and

• the movement from B to C
  remains compatible with the potential that A originally stabilised.

Composition becomes ecological continuity.

The associativity law becomes:

“multi-step perspectival shifts must maintain horizon viability.”

Nothing abstract.

Pure relational constraint.

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5. Categories as Relational Ecologies

If objects are cuts

and morphisms are actualisation pathways

and identities are persistence conditions

and composition is horizon coherence

then a category is:

a landscape of viable perspectival transformations.

A field of relational moves.

A structured ecology of readiness and constraint.

Not a set of objects with arrows on them.

A relationally stratified potential.

A category is a structured horizon.

This is where the ontology and the mathematics meet.

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6. The Consequence: No More “Substances with Structure”

This cut immediately eliminates several inherited metaphysical assumptions:

• No objects with internal content

• No morphisms as “functions”

• No space of entities waiting to be observed

• No compositionality without energetic or perspectival cost

• No algebra without ecology

The category is not a formalism.

It is a relational dynamics.

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7. Why Begin Here?

Because beginning with relation, not object, does three things:

1. It breaks with substance metaphysics entirely.
   Category theory becomes the mathematics of cuts, not things.

2. It grounds the abstract machinery in relational ontology.
   This provides a unified metaphysical base.

3. It prepares the way for the deeper recuts:

   • functors as cross-scale horizon translations

   • natural transformations as compatibility conditions between construal strategies

   • limits and colimits as perspectival stabilisations

   • adjunctions as dual readiness regimes

   • monads as horizon-thickenings

   • toposes (topoi) as full ecological meaning-arenas

This post makes all of that possible.

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Post 1 closes with the thesis:

Category theory, taken relationally, is not abstract mathematics.

It is the formal grammar of how horizons transform.

When the object disappears into the cut,

category theory stops describing structure

and starts describing possibility.