Document Understanding, Measurement, and Manipulation Using Category Theory

TL;DR

This work introduces a category-theoretic framework to represent and manipulate documents by modeling them as categories of question-answer (QA) pairs, enabling semantic discretization of meaning and information-theoretic analysis. It defines equivalence between assertions and QA pairs, builds a category of QA-equivalence classes, and develops a metric and orthogonalization procedure to extract atomic information components. Through abstractive DAGs and lattice-structured exegesis, the approach yields hierarchical summaries and coherent extensions, with rate-distortion theory providing a principled way to compare summarization performance. The framework is extended to multimodal and probabilistic settings, integrates self-supervised feedback via RLVR, and envisions applications in semantic retrieval, document merging, and prompt engineering under a unified, mathematically grounded paradigm.

Abstract

We apply category theory to extract multimodal document structure which leads us to develop information theoretic measures, content summarization and extension, and self-supervised improvement of large pretrained models. We first develop a mathematical representation of a document as a category of question-answer pairs. Second, we develop an orthogonalization procedure to divide the information contained in one or more documents into non-overlapping pieces. The structures extracted in the first and second steps lead us to develop methods to measure and enumerate the information contained in a document. We also build on those steps to develop new summarization techniques, as well as to develop a solution to a new problem viz. exegesis resulting in an extension of the original document. Our question-answer pair methodology enables a novel rate distortion analysis of summarization techniques. We implement our techniques using large pretrained models, and we propose a multimodal extension of our overall mathematical framework. Finally, we develop a novel self-supervised method using RLVR to improve large pretrained models using consistency constraints such as composability and closure under certain operations that stem naturally from our category theoretic framework.

Figures (5)

• Figure 1: Two equivalent representations of non-overlapping components of three assertions $A$, $B$, and $C$. On the left is a symmetric representation; on the right is a representation generated from $A$ and $B$ with $C$ then added.
• Figure 2: An example of a DAG of orthogonalized QAs. Here we represent the QAs using their core assertions. Each node represents a node of the abstractive DAG, with its original assertion on the top line. The following lines are the decomposed, orthogonal QAs. This DAG was created from the following sentences: "The integration of digital tools in education has grown rapidly over the past decade. Many teachers now use online platforms to assign homework and track student progress. Schools should invest more in digital learning infrastructure to enhance educational outcomes."
• Figure 3: An example of rate distortion curves comparing two summarization methods. The rate is the number of words in the summary, and the distortion is the percentage of questions answered incorrectly. The dashed lines connect the raw datapoints and the solid lines form the operational rate distortion curve, which uses the lowest distortion method at the given rate or lower.
• Figure 4: A topological space with open sets. A pre-sheaf adds information to each open set. If the information of the pre-sheaf matches on the overlap of open sets, then the pre-sheaf is actually a sheaf.
• Figure 5: The process by which alignment between commander intent and actions on the ground can be evaluated using our multi-modal category theoretic framework.