Category Cuts: 5 Where Possibility Condenses: Limits, Colimits, and the Shape of Constraint

Adjunctions showed us how perspectives can co-individuate without collapsing into one another. But they also revealed something more unsettling: not all relational configurations stabilise. Some arrangements of cuts hold; others shear apart.

To understand why, we must look not at perspectives themselves, but at how constraints gather.

In category theory, this gathering is formalised as limits and colimits. Ontologically, they are not constructions imposed on a world already divided into objects. They are modes by which possibility condenses into coherence—or fails to do so.

From Relations to Condensation

So far, the series has treated categories as theories of possible instances, functors as perspectival shifts, and adjunctions as structures of co-individuation. Each of these operates at the level of relation.

Limits and colimits operate at a different level:

they describe what happens when many relations attempt to stabilise together.

The question they answer is not:

How does one perspective take up another?

but:

When do multiple constraints jointly determine a coherent cut?

Limits: Holding Together

A limit can be read as a point of maximal compatibility: a way of cutting such that all relevant constraints are simultaneously satisfied.

Relationally:

• multiple perspectives impose conditions,

• each condition restricts how the phenomenon can be construed,

• the limit is the construal that satisfies them all—if such a construal exists.

Crucially, limits are not averages or compromises. They are invariant structures:

• change any constraint, and the limit shifts or disappears,

• remove one relation, and the condensation may dissolve.

Ontologically, a limit is not an object.

It is a pattern of stability in the space of possible instantiations.

When Limits Fail

Not all systems admit limits.

Sometimes:

• constraints conflict irreducibly,

• perspectives demand incompatible cuts,

• no single construal can satisfy all relations at once.

This is not a failure of description.

It is a structural impossibility.

In Impossible Horizons, impossibility marked the productive edge of possibility. Here, we can say more precisely why impossibility arises: the relational constraints do not converge.

The absence of a limit is itself meaningful—not as content, but as structural signal.

Colimits: Opening Out

If limits describe condensation, colimits describe expansion.

Where limits ask:

How can constraints hold together?

colimits ask:

How can differences be jointly accommodated without erasure?

A colimit gathers multiple perspectives by allowing them to remain distinct, while still participating in a shared structure.

Relationally:

• different cuts are preserved,

• incompatibilities are not resolved,

• but a higher-order configuration emerges that holds them in relation.

This is not synthesis.

It is co-presence under constraint.

Emergence Without Fusion

Colimits are especially important for understanding emergence.

Emergent structures are often mystified as “more than the sum of their parts.” From a relational perspective, this misses the point. What matters is not addition, but configuration.

A colimit shows how:

• new forms can actualise,

• without reducing earlier distinctions,

• without requiring prior unity.

Emergence, here, is the actualisation of a new cut in possibility-space, not the production of a new substance.

Meaning, Carefully

This framework lets us speak about meaning without conflating it with value or coordination.

Meaning systems condense where:

• symbolic distinctions,

• contextual constraints,

• and perspectival relations

either admit a limit (tight stabilisation) or require a colimit (structured plurality).

Value systems may coordinate behaviour.

Meaning systems organise construal.

Limits and colimits describe the latter—without importing teleology, function, or normativity.

The Geometry of Possibility

Taken together, limits and colimits give us a geometry of possibility:

• some regions condense tightly,

• some remain irreducibly plural,

• some refuse to stabilise at all.

This geometry is not static.

As constraints evolve, the shape of the space changes.

Which brings us to the final turn of this sequence:

what happens when the theory of cuts itself becomes a site of variation?

The next post will address higher-order cuts—meta-categories, reflexive constraints, and the evolution of the very space in which possibility is articulated.

The horizon is no longer merely open.

It is now folded back on itself.