Relation (multi-source admissible morphism)

A relation, in the τ-framework, is an admissible morphism (or admissible diagram) connecting multiple particulars to a shared structural position — either a universal or a colimit-object. Binary relations are pair-source morphisms; n-ary relations are diagrams of arity n. Per the relational primacy principle (VII.D23), relations are ontologically prior to relata: a particular's identity is given by its relational position, not by relata-independent substance.

Categorical invariant. R is an n-ary relation among (a₁,…,aₙ) ⟺ ∃ admissible diagram D : {aᵢ} → U_R in K_τ.

Primary registry anchor: VII.D23

Supporting items: VII.D40, VII.D36, VII.D25

1. VII.D25 — Internal Set Ontology — relata and relation-targets are NF-addressable.
2. VII.D23 — Relational Primacy — relations precede relata; identity is determined by relational position, not intrinsic substance.
3. VII.D36 — Abstract Object as Structural Position — relation-targets are positions; n-ary relations are admissible diagrams targeting them.
4. VII.D40 — Non-Dualistic Platonism — relations are morphisms (or diagrams of morphisms), not a separate ontological tier.

Lean modules referenced: TauLib.BookVII.Meta.Registers

A relation is instantiated wherever multiple subjects share a connecting predicate: 'a is taller than b', 'a is parent of b and c', 'this lemma supports those theorems', 'this commitment binds these agents'. Each multi-place predication is an admissible diagram in the appropriate register's morphism-class. Relational primacy means the relata are constituted by their relational position — there are no relata-independent particulars waiting to enter relations.

Examples:

• Empirical: 'a is taller than b' — admissible binary diagram in Reg_E targeting the taller-than relation-object, with a, b as relata.
• Genealogical: 'a is parent of b and c' — admissible ternary diagram in Reg_E (or Reg_P, depending on the legal/biological framing) with a, b, c as relata.
• Proof-theoretic: 'this lemma supports those theorems' — admissible diagram in Reg_D from the lemma to each supported theorem.
• Commitment-theoretic: 'this contract binds these signatories' — admissible diagram in Reg_C from contract-content to each signatory's stance.

Register codomain: Cross-register (relations are register-relative; the diagram structure is uniform).

Manuscript reference: manuscript-sources/book-07/part02/ch16.tex

Status: Formalized

Module: TauLib.BookVII.Meta.Registers

Lean kind: structure

Lean symbol: Tau.BookVII.Meta.Registers.NonDualisticPlatonism

Related glossary entries

• MG-O09-property Property (instantiation morphism)
• MG-O07-particular Particular (instantiating source)
• MG-O06-universal Universal (structural position)

Referenced by

• MG-O09-property Property (instantiation morphism)