Yoneda-as-theorem under self-enrichment
Yoneda-as-theorem (II.D50) is the τ-framework's distinctive treatment of the Yoneda lemma: rather than appearing as an external categorical fact applied to τ, the Yoneda embedding is internalized as a theorem provable inside τ via self-enrichment. The construction is the keystone of Book II — it is what makes the Yoneda enrichment ladder (T05 / II.T36) and the central theorem (T04 / II.T40) possible.
τ-Definition
Yoneda-as-theorem (II.D50) is the τ-framework's distinctive treatment of the Yoneda lemma: rather than appearing as an external categorical fact applied to τ, the Yoneda embedding is internalized as a theorem provable inside τ via self-enrichment. The construction is the keystone of Book II — it is what makes the Yoneda enrichment ladder (T05 / II.T36) and the central theorem (T04 / II.T40) possible.
Categorical invariant. The pre-Yoneda embedding Y : τ → [τ^op, τ-Set] viewed as a τ-internal arrow under the self-enrichment of τ; combined with self-enrichment, the embedding becomes a τ-internal theorem rather than an external structural fact.
Primary registry anchor:
II.D50
τ-Derivation Chain
-
I.K0— Universe Postulate -
II.D53— Self-enrichment structure — τ is enriched over itself -
II.D50— Pre-Yoneda embedding — Y : τ → [τ^op, τ-Set] as a τ-internal arrow -
II.T36— Yoneda enrichment ladder — Y extends to a full enrichment, making the lemma a theorem about τ
Lean modules referenced:
TauLib.BookII.Enrichment.YonedaTheorem,
TauLib.BookII.Enrichment.SelfEnrichment
Mathematical content
Define the pre-Yoneda embedding Y : τ → [τ^op, τ-Set] sending each object A to the representable presheaf hom_τ(-, A). Under self-enrichment of τ, Y lifts to a τ-internal functor; the Yoneda lemma — that Y is fully faithful — becomes a theorem provable inside τ.
Novelty. Conventional category theory states the Yoneda lemma as a meta-theorem about functors out of a category C. The τ-framework, exploiting self-enrichment, restates the lemma as an internal theorem about τ itself. This is what 'Yoneda-as-theorem' means: the lemma is no longer a slogan applied to τ, but a result proved in τ.
Consequence. Yoneda-as-theorem is the keystone of Book II: it makes the enrichment ladder (T05) and the central theorem (T04) possible. Without internalization, both theorems would require external Mathlib machinery; with internalization, they are τ-categorical.
Lean Coverage
Status: Formalized
Module: TauLib.BookII.Enrichment.YonedaTheorem
Lean kind: theorem
Lean symbol: Tau.BookII.Enrichment.yonedaAsTheorem