# Sheaf theory: from deep geometry to deep learning

arXiv: https://arxiv.org/abs/2502.15476 · HTML: https://mlsanity.com/paper/2502.15476/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2025.

## Built on by (most cited first, 10 of 10)

- [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://mlsanity.com/paper/2505.21251/) (2025)
- [Learning the Structure of Connection Graphs](https://mlsanity.com/paper/2510.11245/) (2025)
- [Sheaf-Theoretic Transport and Obstruction for Detecting Scientific Theory Shift in AI Agents](https://mlsanity.com/paper/2605.14033/) (2026)
- [On the Sheafification of Higher-Order Message Passing](https://mlsanity.com/paper/2509.23020/) (2025)
- [Learning Latent Graph Geometry via Fixed-Point Schr"odinger-Type Activation: A Theoretical Study](https://mlsanity.com/paper/2507.20088/) (2025)
- [Cycle is All You Need: More Is Different](https://mlsanity.com/paper/2509.21340/) (2025)
- [SheafStain: Sheaf-Theoretic Schr"odinger Bridge for Spatially and Biologically Coherent Virtual Staining](https://mlsanity.com/paper/2606.11846/) (2026)
- [Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling](https://mlsanity.com/paper/2608.01318/) (2026)
- [Capability Sheaves for Compositional Agent-Harness Repair: Controlled Quotients and a Real-Repository Stress Test](https://mlsanity.com/paper/2608.13228/) (2026)
- [Global Coherence: When Every Agent Is Right and the Team Is Still Wrong - A Local-to-Global Semantic Foundation for Multi-Agent Collaboration](https://mlsanity.com/paper/2610.02036/) (2026)

## Abstract (arXiv)

This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory accessible to those with modest mathematical familiarity. We describe intuitions and motivations underlying sheaf theory shared by both theoretical researchers and practitioners, bridging classical mathematical theory and its more recent implementations within signal processing and deep learning. We observe that most notions commonly considered specific to cellular sheaves translate to sheaves on arbitrary posets, providing an interesting avenue for further generalization of these methods in applications, and we present a new algorithm to compute sheaf cohomology on arbitrary finite posets in response. By integrating classical theory with recent applications, this work reveals certain blind spots in current machine learning practices. We conclude with a list of problems related to sheaf-theoretic applications that we find mathematically insightful and practically instructive to solve. To ensure the exposition of sheaf theory is self-contained, a rigorous mathematical introduction is provided in appendices which moves from an introduction of diagrams and sheaves to the definition of derived functors, higher order cohomology, sheaf Laplacians, sheaf diffusion, and interconnections of these subjects therein.