Category Cuts: 3 Perspectival Shifts: Functors and the Translation of Possibility

If categories are theories of possible instances, then the next problem is unavoidable: no category exists alone. Perspectives do not arise in isolation; they emerge in relation to other perspectives, other systems, other cuts.

The question is not whether different theories of possibility relate.

The question is how they do so without reduction, dominance, or incoherence.

This post introduces functors—not as technical mappings, but as structured perspectival shifts.

The Problem of Translation

Whenever we move from one system of meaning to another—

from physics to language,

from individual experience to social form,

from one theoretical framing to another—

we face a problem of translation.

Naively, translation is treated as:

• representation (“this thing corresponds to that thing”),

• reduction (“this can be explained in terms of that”),

• or metaphor (“this is like that”).

Relational ontology rejects all three as primary.

A perspectival shift does not preserve things.

It preserves structure under constraint.

This is exactly what functors were invented to do.

Functors as Perspectival Cuts

In category theory, a functor maps:

• objects to objects,

• morphisms to morphisms,

• while preserving composition and identity.

Read ontologically, this becomes:

A functor is a cut that takes one theory of possible instances as an instance within another theory, without exhausting either.

This is crucial.

A functor does not claim that two categories are the same.

Nor does it collapse one into the other.

It specifies which distinctions survive the shift of perspective.

What is preserved is not content, but relational intelligibility.

Preservation Is the Ontological Act

Preservation is not a technical convenience.

It is the ontological heart of the matter.

To preserve composition is to say:

If a sequence of cuts is coherent here, it remains coherent there—though possibly reinterpreted.

To preserve identities is to say:

Some stabilisations remain stabilisations across perspectives.

Equally important is what functors do not preserve:

• intensities,

• local salience,

• phenomenological weight,

• or domain-specific value.

This is why functors do not conflate systems. They articulate selective continuity, not sameness.

Against Reduction and Relativism

Seen this way, functors block two persistent errors at once.

They block reductionism, because:

• no functor preserves everything,

• every translation loses and reshapes structure,

• and source and target categories remain distinct.

They also block naive relativism, because:

• not all mappings are functorial,

• not all perspectival shifts preserve coherence,

• and failure of preservation is diagnostically meaningful.

A bad translation is not “just another perspective”.

It is a structural failure.

Functors and Meaning

This matters deeply for meaning.

Meaning does not travel intact across perspectives.

Nor does it dissolve into incommensurability.

Instead, meaning:

• transforms under constraint,

• survives only where relational structure survives,

• and mutates where composition fails.

Functors give us a way to talk about this precisely:

• which distinctions carry,

• which collapse,

• which re-emerge differently.

This is not semantics as correspondence.

It is semantics as structured transfer of possibility.

Multiple Functors, Multiple Readings

Between any two categories, there may be many functors—or none.

This is not a problem. It is a feature.

Multiple functors correspond to:

• different legitimate construals,

• different explanatory priorities,

• different cuts across the same relational terrain.

But not all functors are equal.

Some preserve more structure.

Some reveal new limits.

Some expose hidden tensions.

The evaluation of functors is therefore an ontological practice, not a neutral choice.

From Translation to Co-Individuation

So far, functors describe one-way perspectival shifts.

But many relations are reciprocal, asymmetric, and mutually constraining.

When two systems do not merely translate but co-stabilise each other, we need stronger structure.

This brings us to adjunctions: paired functors that formalise dialogic co-individuation, where each perspective both enables and constrains the other.

That will be the task of the next post.

The horizon remains open.

The cuts now move—not in isolation, but in structured relation.