Chapter 43: τ Enriches Over Itself

In most categories, morphism spaces are external: Hom(A,B) is a set, living in the ambient category Set rather than in the category under study. An enriched category replaces these external sets with internal objects—Hom objects that are themselves objects of the category. This chapter proves that Category τ enriches over itself: for any two objects A, B ∈ Obj(τ), the morphism space Hom_τ(A,B) is itself a τ-object, with an NF-address, split-complex values, and a tower-coherent structure inherited from the primorial tower.

**the relevant definition (II.D53): τ is self-enriched—its composition and identity maps are τ-morphisms. **the relevant definition (II.D54): the Hom object [A,B] is the space of τ-holomorphic maps, equipped with tower-coherent staging [A,B]k = Hom(A_k, B_k). Proposition [prop:hom-bipolar] (II.P11): every Hom object inherits the bipolar decomposition, [A,B] = e+ · [A,B]+ + e- · [A,B]_-, because holomorphic maps between τ-objects are themselves holomorphic objects, and the Idempotent Decomposition Lemma (II.L07) applies at the level of morphism spaces.

The argument rests on the Pre-Yoneda embedding (the relevant definition, II.D50, the relevant chapter): holomorphic functions are objects, so function spaces are objects, so Hom spaces are objects. Self-enrichment is the structural precondition for the Yoneda embedding , and it initiates the transition from E₀⁽0⁾ (Book I + Book II Parts I–VII) to E₀⁽1⁾ (Book II Part VIII).