# Topos Theory for Generative AI and LLMs

arXiv: https://arxiv.org/abs/2508.08293 · HTML: https://mlsanity.com/paper/2508.08293/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 27 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Faith and Fate: Limits of Transformers on Compositionality](https://mlsanity.com/paper/2305.18654/) (2023)
- [Theoretical Limitations of Self-Attention in Neural Sequence Models](https://mlsanity.com/paper/1906.06755/) (2019)
- [Saturated Transformers are Constant-Depth Threshold Circuits](https://mlsanity.com/paper/2106.16213/) (2021)
- [An Attentive Survey of Attention Models](https://mlsanity.com/paper/1904.02874/) (2019)
- [Tighter Bounds on the Expressivity of Transformer Encoders](https://mlsanity.com/paper/2301.10743/) (2023)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https://mlsanity.com/paper/2402.15332/) (2024)
- [An enriched category theory of language: from syntax to semantics](https://mlsanity.com/paper/2106.07890/) (2021)
- [What makes Models Compositional? A Theoretical View: With Supplement](https://mlsanity.com/paper/2405.02350/) (2024)

## Built on by (most cited first, 2 of 2)

- [The Neurosymbolic Frontier of Nonuniform Ellipticity: Formalizing Sharp Schauder Theory via Topos-Theoretic Reasoning Models](https://mlsanity.com/paper/2602.10632/) (2026)
- [The Cognitive Categorical Transformer: Category-Theoretic Inductive Biases for Language Modeling](https://mlsanity.com/paper/2605.28864/) (2026)

## Abstract (arXiv)

We propose the design of novel categorical generative AI architectures (GAIAs) using topos theory, a type of category that is ``set-like": a topos has all (co)limits, is Cartesian closed, and has a subobject classifier. Previous theoretical results on the Transformer model have shown that it is a universal sequence-to-sequence function approximator, and dense in the space of all continuous functions with compact support on the Euclidean space of embeddings of tokens. Building on this theoretical result, we explore novel architectures for LLMs that exploit the property that the category of LLMs, viewed as functions, forms a topos. Previous studies of large language models (LLMs) have focused on daisy-chained linear architectures or mixture-of-experts. In this paper, we use universal constructions in category theory to construct novel LLM architectures based on new types of compositional structures. In particular, these new compositional structures are derived from universal properties of LLM categories, and include pullback, pushout, (co) equalizers, exponential objects, and subobject classifiers. We theoretically validate these new compositional structures by showing that the category of LLMs is (co)complete, meaning that all diagrams have solutions in the form of (co)limits. Building on this completeness result, we then show that the category of LLMs forms a topos, a ``set-like" category, which requires showing the existence of exponential objects as well as subobject classifiers. We use a functorial characterization of backpropagation to define a potential implementation of an LLM topos architecture.