# Weaves, Wires, and Morphisms: Formalizing and Implementing the Algebra of Deep Learning

arXiv: https://arxiv.org/abs/2604.07242 · HTML: https://mlsanity.com/paper/2604.07242/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 36 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [FlashAttention-3: Fast and Accurate Attention with Asynchrony and Low-precision](https://mlsanity.com/paper/2407.08608/) (2024)
- [AI and Memory Wall](https://mlsanity.com/paper/2403.14123/) (2024)
- [Formal Algorithms for Transformers](https://mlsanity.com/paper/2207.09238/) (2022)
- [FlashAttention-4: Algorithm and Kernel Pipelining Co-Design for Asymmetric Hardware Scaling](https://mlsanity.com/paper/2603.05451/) (2026)
- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https://mlsanity.com/paper/2402.15332/) (2024)
- [Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability](https://mlsanity.com/paper/2010.07416/) (2020)
- [Named Tensor Notation](https://mlsanity.com/paper/2102.13196/) (2021)
- [An Introduction to String Diagrams for Computer Scientists](https://mlsanity.com/paper/2305.08768/) (2023)
- [Neural Circuit Diagrams: Robust Diagrams for the Communication, Implementation, and Analysis of Deep Learning Architectures](https://mlsanity.com/paper/2402.05424/) (2024)
- [FlashAttention on a Napkin: A Diagrammatic Approach to Deep Learning IO-Awareness](https://mlsanity.com/paper/2412.03317/) (2024)
- [Fundamental Components of Deep Learning: A category-theoretic approach](https://mlsanity.com/paper/2403.13001/) (2024)
- [Deep Learning with Parametric Lenses](https://mlsanity.com/paper/2404.00408/) (2024)
- [Accelerating Machine Learning Systems via Category Theory: Applications to Spherical Attention for Gene Regulatory Networks](https://mlsanity.com/paper/2505.09326/) (2025)

## Abstract (arXiv)

Despite deep learning models running well-defined mathematical functions, we lack a formal mathematical framework for describing model architectures. Ad-hoc notation, diagrams, and pseudocode poorly handle nonlinear broadcasting and the relationship between individual components and composed models. This paper introduces a categorical framework for deep learning models that formalizes broadcasting through the novel axis-stride and array-broadcasted categories. This allows the mathematical function underlying architectures to be precisely expressed and manipulated in a compositional manner. These mathematical definitions are translated into human manageable diagrams and machine manageable data structures. We provide a mirrored implementation in Python (pyncd) and TypeScript (tsncd) to show the universal aspect of our framework, along with features including algebraic construction, graph conversion, PyTorch compilation and diagram rendering. This lays the foundation for a systematic, formal approach to deep learning model design and analysis.