Yoneda enrichment ladder
The Yoneda enrichment ladder (II.T36) is the Book-II theorem that lifts the pre-Yoneda embedding (D04 / II.D50) to a full self-enrichment of τ. The ladder exhibits a sequence of enrichment steps Y_0 ⊆ Y_1 ⊆ Y_2 ⊆ … each refining the previous, with the limit equal to the full Yoneda-as-theorem statement. It is the constructive proof of D04: not just that Yoneda is internalizable, but that the internalization is reachable as the limit of an explicit chain.
τ-Definition
The Yoneda enrichment ladder (II.T36) is the Book-II theorem that lifts the pre-Yoneda embedding (D04 / II.D50) to a full self-enrichment of τ. The ladder exhibits a sequence of enrichment steps Y_0 ⊆ Y_1 ⊆ Y_2 ⊆ … each refining the previous, with the limit equal to the full Yoneda-as-theorem statement. It is the constructive proof of D04: not just that Yoneda is internalizable, but that the internalization is reachable as the limit of an explicit chain.
Categorical invariant. An ω-chain of enrichment refinements on the pre-Yoneda embedding, whose colimit is the full self-enriched τ-internal Yoneda functor.
Primary registry anchor:
II.T36
τ-Derivation Chain
-
I.K0— Universe Postulate -
II.D50— Pre-Yoneda embedding Y : τ → [τ^op, τ-Set] -
II.D53— Self-enrichment structure on τ -
II.T36— Yoneda enrichment ladder — ω-chain Y_0 ⊆ Y_1 ⊆ … with colimit = full self-enriched Yoneda
Lean modules referenced:
TauLib.BookII.Enrichment.EnrichmentLadder,
TauLib.BookII.Enrichment.YonedaTheorem
Mathematical content
There exists an ω-chain Y_0 ⊆ Y_1 ⊆ Y_2 ⊆ … of enrichments on the pre-Yoneda embedding (D04 / II.D50), each refining the previous by one Yoneda-as-theorem application step. The colimit of the chain equals the full self-enriched τ-internal Yoneda functor — i.e., the full Yoneda-as-theorem of D04.
Constructive aspect. The ladder gives a constructive proof of D04: not just that Yoneda-internalization exists, but that it is reachable as the limit of an explicit, computable chain. Each rung Y_n is a τ-internal functor; the chain is well-founded and terminates at ω.
Consequence. T05 is the input to the Central theorem (T04 / II.T40): the categoricity check at rank (3, 15) needs the enrichment ladder to have closed (i.e., the ω-chain to have reached its colimit) before the algebraic invariant is well-defined.
Lean Coverage
Status: Formalized
Module: TauLib.BookII.Enrichment.EnrichmentLadder
Lean kind: theorem
Lean symbol: Tau.BookII.Enrichment.yonedaLadder