Category Theory Through the Lens of Relational Ontology: 6 Bicategories and the Softness of Systemic Horizons

Why Not All Horizons Are Rigid — and Why This Matters for Relation

In the previous post, we saw how higher-order relationality arises inevitably once construal itself acquires horizon-like structure.
But there is a subtlety—an ontological inflection point—hidden in the transition from categories to bicategories:

Not all horizons hold their boundaries with the same rigidity.
Some horizons flex.
Some bend.
Some maintain coherence without enforcing strict equalities.

Category theory formalises this as the shift from:

• strict identity laws
  

  to

• up-to-isomorphism coherence.

Relational ontology frames the same shift as:

• the softness of ecological horizons,

• the tolerance inherent in lived relationality,

• the non-crisp nature of multi-scale alignment,

• the fact that many relational structures persist without exactitude.

This post introduces bicategories not as mathematical oddities, but as natural expressions of soft relational ecology.

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1. The Problem: Strict Horizons Are Too Clean

In an ordinary (strict) category, identities and compositions must satisfy rigid equalities:

$$(f \circ g) \circ h = f \circ (g \circ h)$$

This is elegant from a formal standpoint—but ontologically brittle.
Real horizons, biological horizons, cognitive horizons—none of them operate on strict associativity.

Nature does not say:

“This pathway composed with that pathway is literally identical to the other grouping.”

Instead:

It maintains coherence up to equivalence.
Systems align “close enough” to function.

This is not sloppiness.
It is the structural condition of a relational world where no horizon is frozen.

Thus the strict category is too rigid to model relational ontology in full fidelity.

Something more supple is needed.

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2. Bicategories: Coherence Instead of Equality

A bicategory relaxes the rigid equalities but preserves coherence via isomorphisms rather than equations.

Instead of demanding:

$$(f \circ g) \circ h = f \circ (g \circ h)$$

we require only that there exists a coherent associator:

$$\alpha_{f,g,h} : (f \circ g) \circ h \Rightarrow f \circ (g \circ h)$$

a natural, invertible 2-morphism that witnesses the alignment between the two composites.

Similarly, identity morphisms are replaced by unitors—coherent isomorphisms that align self-relations with larger pathways.

From a relational perspective, this expresses a simple truth:

Horizons do not produce identical pathways;
they produce coherent families of pathways.

A bicategory is thus a mathematics of soft horizons.

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3. Soft Horizons: The Relational Interpretation

In relational ontology, horizons are not static.
They dynamically shift with:

• metabolic readiness

• ecological coupling

• multi-scale attunement

• changes of perspective

• construal-level integration

Therefore:

Alignment across horizons is rarely exact.
It is negotiated, modulated, and “good enough” for action.

This is exactly what bicategories capture.

Strict categories

model rigid representational universes.

Bicategories

model ecologies of relational potential where:

• boundaries deform,

• mappings flex,

• and coherence is preserved without requiring identity.

They are mathematically dignified versions of biological and cognitive adaptability.

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4. Associators as Modulations of Flexibility

The associator $\alpha_{f,g,h}$ is not an algebraic leftover.

Ontologically, it expresses:

The modulation needed to transport relational structure across different perspectives on the same composite pathway.

Whenever we chain processes in a living or cognitive system—perception → construal → action—there is:

• a soft negotiation of relevance,

• a reconfiguration of readiness,

• a reselection of gradient sensitivity.

The associator is the formal trace of that modulation.

It is what ensures that even though the horizon flexes, the system’s coherence is not lost.

This is exactly how cognition maintains stability without rigidity.

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5. Bicategories as Models of Meaningful Practice

Every complex semiotic system—language, culture, mind—operates more like a bicategory than a strict one.

Why?

Because meaning is never:

• fixed,

• rigidly compositional,

• or exact in its pathways.

Meaning is:

• systemic but deformable,

• coherent but perspectival,

• stable enough to coordinate action,

• flexible enough to accommodate context and stratification.

A bicategory is the cleanest formal analogue of this.

This is the point where category theory starts to explicitly mirror the Hallidayan insight that meaning systems are:

• patterned,

• systemic,

• relational,
  but also

• inherently soft,

• inherently social,

• inherently contextually modulated.

The mathematics finally meets the semiotics without collapsing into representation.

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6. The Philosophical Consequence: Strict Systems Are the Exception, Not the Rule

Most of mathematics prefers strict categories because they are easier.
But ontologically:

Strictness is a degenerate case of relational ecology.
Soft coherence is the norm.

This flips the typical metaphysical orientation:

• Instead of thinking bicategories “generalise” categories,
  we see categories as artificially rigid caricatures of a richer relational field.

• Instead of thinking associators compensate for failure of equality,
  we see equalities as suppressions of relational nuance.

• Instead of thinking bicategories are complicated,
  we see them as minimal realism.

This is a philosophical realignment:
the world is bicategorical; strictness is the abstraction.

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7. Toward Post 7: Higher Softness, Higher Ecology

Once we accept that:

• horizons are soft,

• alignments are up-to-isomorphism,

• coherence replaces equality,

we arrive naturally at the final question:

What does a fully relational ontology look like when the entire multi-scale hierarchy is soft, perspectival, coherent, and endlessly open to further construal?

Category theory answers with:

• tricategories,

• $(\infty, n)$-categories,

• fully weak higher structures.

Relational ontology answers with:

• a vision of reality as an ecology of actualisations,

• coherence rather than identity,

• alignment rather than representation,

• and meaning as cross-scale relational attunement.

This is Post 7, the capstone:
When the Hierarchy Never Ends: Infinity-Categories and the Open Relational Cosmos.