Typal Meta-Operational Mathematics: From Iteration of Typal Operations to Operations on Typal Operations

This work develops Typal Meta-Operational Mathematics, a systematic framework that elevates typal operations-type constructors, dependent functions, higher inductive types, identity types, and their inverses-to the status of independent mathematical objects. Unlike previous meta-operational theories where a set of axioms is postulated, we build the entire theory on the firm ground of standard ZFC set theory with a Grothendieck universe and the established machinery of ∞-category theory. We prove that the collection of all typal operations forms a coloured symmetric operad TypeOp, which is further endowed with a Hopf operad structure. In this structure, adjoint dualities play the role of the antipode, and the triangular identities of adjunctions are equivalent to the antipode axioms. The antipode is globally extended to all type constructors via homotopy bornological closure, and we establish its compatibility with Kan extensions and derived constructions. Acomplete classification of higher-order antipodes is proved: every antipode is of exactly one of three types-involution (Z2), order-4 cyclic (Z4), or continuous flow (U(1)-family). A concrete Hopf algebra morphism from the primitive algebra of unary typal meta-operations to a categorified Connes-Kreimer renormalization Hopf algebra is constructed, with a faithful graphical representation via typal Feynman diagrams. The non-idempotence spectrum is introduced and computed in concrete examples, revealing phase transitions and collapse phenomena. Weighted parametrized families of type constructors are studied, and collapse thresholds are identified as critical boundaries of dynamic phase transitions. All classical homotopy type theory identities-Yoneda lemma, triangular identities, monad and comonad laws, Kan extension universal property, and the univalence axiom-are expressed as meta-operational equations. The entire framework is categorified into a strict 2-category and further lifted to an ∞-operad TypeOp∞, in which a univalent universe is realized.