Token Space: A Category Theory Framework for AI Computations

arXiv:2404.11624v4 Announce Type: replace-cross Abstract: Token Space is a categorical framework and mathematical language for AI computations. It connects internal relations, program descriptions, execution states and observations, locating questions about structure, behavior and cost at their appropriate levels. Five guiding positions concern structural interiors, categorical self-description, interfaces, occurrence identity and extensible computation. Tokens are finite records of carrier elements and fixed symbols; Token maps preserve selected heaps. The elementary category is a quasitopos, hence locally cartesian closed, but not a topos. Algebraic tokenization is fully faithful for fixed finitary signatures. Represented finite mappings admit concurrent graph execution, gluing, functorial frontiers and exact state migration characterized by kernel inclusion. Transformers are one implementation family. Coherent occurrence prefixes yield natural numerical maps, and a cache invariant proves agreement with full-prefix evaluation. A prescribed access policy determines the least retained index set under a no-reconstruction discipline. Parameterised state expresses changing interfaces. For a specified future-observation heap, structural indiscernibility is behavioral equivalence. An encoding admits exact incremental execution precisely when its kernel is a right congruence contained in that equivalence; every reachable exact realization maps uniquely onto the behavioral quotient. Teacher-induced heaps and declared readouts connect these constructions to distillation and explicit knowledge. The definitions, examples and theorems demonstrate how the language joins lines of reasoning while distinguishing representation, execution and observation. Effective implementations and quantitative performance remain further questions.