Compositional Algorithms with Sheaves

Four pictures of the same idea: overlapping open sets on a space, the three-colorings of a graph, vector spaces stacked over a graph's vertices and edges, and the solutions of a PDE patched across a grid
Sheaves in four settings, each with data attached locally and agreement conditions where the pieces overlap. From the compositional methods talk.

StructuredDecompositions.jl

Project team

Project articles

  1. Bumpus, B. M., Leal, W., Fairbanks, J., Karvonen, M., & Simard, F. (2026). Towards a Unified Theory of Time-Varying Data. Applied Categorical Structures, 34(3), 23. https://doi.org/10.1007/s10485-026-09860-4
  2. Azevedo, A. B., Bumpus, B. M., Capucci, M., Fairbanks, J., & Rosiak, D. (2025). Algorithmic and Extremal Obstructions Through the Language of Cohomology. arXiv. https://doi.org/10.48550/arXiv.2407.03488
  3. Bumpus, B. M., Fairbanks, J., & Turner, W. J. (2024). Pushing Tree Decompositions Forward Along Graph Homomorphisms. arXiv. https://doi.org/10.48550/arXiv.2408.15184
  4. Bumpus, B. M., Fairbanks, J., Genovese, F., Puca, C., & Rosiak, D. (2024). How nice is this functor? Two squares and some homology go a long way. Proceedings of Applied Category Theory, 2024.
  5. Bumpus, B. M., Capucci, M., Fairbanks, J., & Rosiak, D. (2024). Failures of compositionality: a short note on cohomology, sheafification and lavish presheaves. arXiv:2407.03488.
  6. Althaus, E., Bumpus, B. M., Fairbanks, J., & Rosiak, D. (2023). Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves. arXiv. https://doi.org/10.48550/arXiv.2302.05575

Sponsors

DARPA

Computational sheaf theory has been supported by the DARPA program Automating Scientific Knowledge Extraction and Modeling.