Category Theory Through the Lens of Relational Ontology: 3 Functors as Cross-Scale Translations of the Relational Horizon

Category theory is frequently introduced as the mathematics of structure-preserving mappings. But if we take that phrase literally, we end up missing the point. A functor does not “preserve” structure in any naïve representational sense; it performs—and thereby actualises—a relation between distinct horizons of construal. A functor is not a bridge between two already-given universes of discourse. It is the operation that brings each universe into view as a universe relative to the other.

In relational-ontological terms:

• A category is a perspectival totality—a horizon of possible relational cuts.

• A functor is a cross-scalar operation that re-situates that horizon within another, enacting a translation between potentials.

This translation is not primarily about transporting objects across domains; it is about aligning possibility spaces. A functor maps the theory of an instance at one horizon into an alternative construal at another. What becomes visible under one cut may be re-configured, simplified, or enriched under another. The functor is the device that makes these correspondences legible—indeed, thinkable.

1. Categories as Theories of Relational Potential

Within relational ontology, a category can be construed as a systemic horizon:

• Objects are loci of possible instantiation;

• Morphisms are structured pathways for taking one perspectival cut into another;

• Composition is the stability condition that anchors the horizon as a coherent space of potential transitions.

This treats the category not as a catalogue of entities but as a schema of admissible construals. The horizon is what makes objects intelligible as objects and morphisms intelligible as transformations. Without this horizon, nothing “in” the category exists. It is the totality of relational possibilities that actualises them.

2. What a Functor Actually Does

When a functor 

$F: \mathcal{C} \to \mathcal{D}$

1. Positions one horizon within another;

2. Reinterprets relational potential in a new space of abstractability;

3. Construes the morphic pathways of the source category as meaningful transitions within the target horizon.

A functor is thus a cross-scale interpreter.

It is not translating content; it is translating conditions for the possibility of content.

It is not matching items; it is matching ways of taking relational cuts.

This is why the preservation conditions matter. Preserving identities and composition is not a constraint imposed from outside; it is the condition under which two horizons can enter a relationship that is not merely metaphorical but systemically intelligible.

3. Cross-Scale Translation

The notion of “scale” here is not geometric but epistemic and semiotic.

Each category carries a stance on:

• how granular or abstract a construal is,

• what counts as a stable relation,

• what invariances matter.

A functor translates between these scales by offering a re-individuation of potential: it says, in effect, “If this counts as an object here, then this will count as its aligned potential there.”

Cross-scale translation is therefore a form of meaning-projection: the functor traces which parts of the systemic horizon survive a perspectival shift, and which parts are re-expressed at a different resolution.

4. Functors as Instruments of Relational Alignment

Under this reading, functors are not primarily mathematical gadgets but ontological operators. They articulate how one horizon of potential relates to another. They reveal the mutual hospitality between systemic spaces—what can be taken across without distortion, what must be transformed, what cannot cross at all.

This also explains why natural transformations matter: they express the intra-horizon modulation of these translations, the ways two functors differ not in the structure they “carry” but in the perspective they enact on that structure. But that is the focus of Post 4.

5. Why the Functor Matters for Relational Ontology

A relational ontology committed to systemic potential rather than pre-given objects faces a recurring challenge:

How do we understand the comparability of horizons that are themselves construed?

The functor provides the answer.

It is the formal embodiment of perspectival compatibility. It shows how horizons can be co-individuated: not collapsed into each other, but aligned through the disciplined operational constraints of mapping identities and compositions.

The functor is therefore a model of the cross-horizon cut—a translation that does not reduce but re-situates, enacting meaning across difference.