Self-enrichment construction
The self-enrichment construction (II.D53) equips τ with the structure of a category enriched over itself. Hom-sets in τ become τ-internal objects rather than external Set-elements, and statements about them become statements provable inside τ. The construction is the structural prerequisite for the Yoneda-as-theorem internalization (D04 / II.D50) and the enrichment ladder (T05 / II.T36) — without self-enrichment, the Yoneda lemma stays external and the Central theorem (T04) cannot be τ-categorical.
τ-Definition
The self-enrichment construction (II.D53) equips τ with the structure of a category enriched over itself. Hom-sets in τ become τ-internal objects rather than external Set-elements, and statements about them become statements provable inside τ. The construction is the structural prerequisite for the Yoneda-as-theorem internalization (D04 / II.D50) and the enrichment ladder (T05 / II.T36) — without self-enrichment, the Yoneda lemma stays external and the Central theorem (T04) cannot be τ-categorical.
Categorical invariant. A τ-Cat-enriched-category structure on τ — i.e., the data making τ a category enriched over itself rather than over Set.
Primary registry anchor:
II.D53
τ-Derivation Chain
-
I.K0— Universe Postulate -
II.D53— Self-enrichment construction — τ as a category enriched over itself -
II.D50— Pre-Yoneda embedding lifts via self-enrichment -
II.T36— Yoneda enrichment ladder converges via self-enrichment colimit
Lean modules referenced:
TauLib.BookII.Enrichment.SelfEnrichment,
TauLib.BookII.Enrichment.SelfDescribing,
TauLib.BookII.Enrichment.TwoCategories
Mathematical content
τ is enriched over τ-Cat (the τ-internal 2-category of τ-categories): the hom-sets hom_τ(A, B) are τ-internal objects rather than elements of Set, and composition is a τ-internal morphism rather than an external function.
Construction. The self-enrichment is constructed by exhibiting τ as a Cat-enriched category in the τ-internal 2-category τ-Cat, with hom-objects given by the standard categorical-internal hom and composition inherited from K3.
Consequences:
- The pre-Yoneda embedding Y : τ → [τ^op, τ-Set] lifts to a τ-internal functor.
- The Yoneda lemma — that Y is fully faithful — becomes a τ-internal theorem (Yoneda-as-theorem, D04).
- The enrichment ladder (T05) converges by sequential self-application of the lift step.
Lean Coverage
Status: Formalized
Module: TauLib.BookII.Enrichment.SelfEnrichment
Lean kind: structure
Lean symbol: Tau.BookII.Enrichment.selfEnrichmentStructure