Language, Statistics, & Category Theory, Part 1

Abstraction: Category theory framework unifying algebraic and statistical structure in language

Key points:

• Language is algebraic (compositional concatenation) and statistical (word co-occurrence probabilities); ideals capture only the algebraic part
• Language is modeled as a preordered category L whose morphisms are substring containment relations
• The Yoneda lemma formalizes Firth's quote "you shall know a word by the company it keeps": a word is determined by all expressions containing it
• The copresheaf L(x,−) maps each word to its representable functor, recovering principal ideals as a special case
• The copresheaf category Set^L is a topos, enabling categorical logic (disjunction via coproducts, etc.) beyond what ideals support
• Part 1 of a series motivated by understanding large language model successes through abstract mathematics

Connections: Math3ma · Category Theory · Natural Language Processing · Large Language Models

Source: https://www.math3ma.com/blog/language-statistics-category-theory-part-1