Pure Mathematics for Theoretical Computer Science

A Functorial Structure is defined as a covariant functor F : C → Set, assigning sets to objects and functions to morphisms, ensuring functoriality. In this paper, we introduce and formally define two new concepts: the HybridFunctorial Structure and the MultiFunctorial Structure. A HybridFunctorial Structure combines two functors on the same category, linked by a natural transformation, ensuring consistent pushforward compatibility. A MultiFunctorial Structure involves multiple functors indexed by a preorder, coherently related via natural transformations, forming compatible families with functorial consistency.

Standard probability theory assigns each event a single real value in [0, 1], satisfying non-negativity, normalization, and countable additivity. Hyper-Probability extends this notion by assigning to each event a set of probability values in [0, 1], thereby capturing multiple independent assessments from diverse sources. Super-HyperProbability further generalizes the framework by mapping events to iterated power sets of [0, 1], modeling hierarchical uncertainty across multiple aggregation levels. In this paper, we formally define the Hyper-Probability Measure and Hyper-Probability Distribution, examine their fundamental properties, and demonstrate how these constructs unify and extend classical probability within the Hyper- and Super-HyperProbability paradigms.