Chapter 34: Composition, Identity, Associativity

The preceding chapters of Part VI built the objects and morphisms of split-complex holomorphy: τ-admissible points as objects, ω-germ transformers as morphisms, the BndLift construction , the Mutual Determination Theorem (the relevant chapter, II.T27), and the evolution operator (the relevant chapter, II.D37). This chapter verifies that these morphisms form a category. Composition of holomorphic maps is defined as stagewise composition of ω-germs (the relevant definition, II.D39). The identity map is the constant family assigning the identity germ at each stage (the relevant definition, II.D40). The central result is *the relevant theorem (II.T29): associativity of composition, which is *not trivial. The proof lifts the program monoid’s associativity (I.P02, Book I) through the tower coherence conditions that every holomorphic map must satisfy. With identity and associativity established, the category HolEnd_τ (the relevant definition, II.D41) is formed: the category of τ-holomorphic endomorphisms, the algebraic skeleton on which Parts VII–IX will build.