Operational Mathematics of Functor Operations: Extending the Iteration Count of Functor Composition and Its Inverse to the Complex Domain

This work systematically transplants the core methodology of Operational Mathematics——the extension of the repetition count of basic operations from natural numbers to integers, rational numbers, real numbers, and ultimately complex numbers——to the realm of functor operations. Functor operations encompass all levels of categorical composition: composition of functors, vertical and horizontal composition of natural transformations, and compositions of higher morphisms in (∞,n)-categories. Each such operation possesses distinctive essential features: partial domains of definition (composability depends on source/target matching), non-commutativity, non-idempotence (except for identity functors), and the fact that inverses exist only for equivalences (weak inverses). We establish a complete axiomatic system of seven independent axioms that capture these features. Integer-order, fractional-order, real-order, and complex-order iterations are rigorously defined. By linearising locally small categories into morphism algebras and completing them to Banach algebras, we employ holomorphic functional calculus to define arbitrary complex powers Ft = exp(tlogF) of a functor F, thereby achieving continuous iteration. Existence and uniqueness theorems for iterative roots at each level are proved, using Schröder’s equation, Abel’s equation, and a Kneser-type construction adapted to Banach algebras. The singularity structure of complex-order functor iteration is analysed in depth, revealing a novel phenomenon of mixed algebraic-logarithmic branch points determined by the torsion properties and Jordan block structure of the spectrum of F. When two eigenvalues have rationally independent logarithms, the branch points accumulate densely on curves, forming natural boundaries of Hausdorff dimension 2 under suitable Diophantine conditions.