arXiv：cs.LG（机器学习，全量分类）· Wuming Pan·· 15 小时前AI 评分24
Token Space：面向 AI 计算的范畴论框架
Token Space: A Category Theory Framework for AI Computations
AI 导读
研究者提出 Token Space，一个基于显式结构记录的 AI 计算范畴论框架，由五条论题引导，将 Token 定义为载体元素与固定符号的有限元组。该框架的基本范畴具有有限极限、有限余积与指数对象，但不是 topos；小范畴与函子可被记录编码，有限范畴构造可执行。Transformer 被视为其中一种实现族，置换堆刻画等变性、前缀一致堆刻画因果性，结构蒸馏则使用教师诱导的堆。
正文
Abstract:We introduce Token Space, a categorical framework for AI computations based on explicit structural records. Five theses guide it: object interiors should be data; category theory should compute with its own objects; computational interfaces should specify structural obligations; equal vectors need not identify the same Token occurrence; and computation should admit an unbounded, dynamically organized population of Token computing cores. A Token is a finite tuple of carrier elements and fixed symbols. A Token class pairs a carrier with a heap of records; Token maps preserve those records. The elementary category has finite limits, finite coproducts and exponentials, but is not a topos. Algebraic tokenization is fully faithful for a fixed finitary signature with all homomorphisms. Small categories and functors have record encodings, natural transformations have endpoint-constrained encodings, and finite categorical constructions are executable. Operators and supported tree reification expose internal structure. For represented finite mappings, valid acyclic graphs evaluate through unique Token maps. Completed parts glue by pullback-pushout squares, sharing induces an adjunction on completion lattices, frontier interfaces form a functor, and certified residual replacement preserves the remaining result. Effective finite transitions preserve finite configurations; a uniform generator yields arbitrarily wide ready populations. Requests with finite dependency closures complete under stated progress conditions. Transformers are one implementation family: permutation heaps characterize equivariance and prefix-agreement heaps characterize causality under specified interfaces. Structural distillation uses teacher-induced heaps; relation-saturating quotients characterize exact preservation and reflection of recorded structure.
| Comments: | 83 pages, 15 figures, 11 tables. Substantially revised and extended: five theses; represented finite-mapping machines; concurrent and elastic execution; unbounded Token computing cores. Validation code and results included as ancillary files |
| Subjects: | General Mathematics (math.GM); Machine Learning (cs.LG) |
| MSC classes: | I.2.6 |
| Cite as: | arXiv:2404.11624 [math.GM] |
| (or arXiv:2404.11624v3 [math.GM] for this version) | |
| https://doi.org/10.48550/arXiv.2404.11624 arXiv-issued DOI via DataCite |
Submission history
From: Wuming Pan [view email]
[v1]
Thu, 11 Apr 2024 15:56:06 UTC (30 KB)
[v2]
Wed, 30 Sep 2026 15:02:17 UTC (45 KB)
[v3]
Thu, 1 Oct 2026 13:10:31 UTC (118 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org