Bringing compositionality to technical computing.
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AlgebraicJulia aims to create novel approaches to scientific computing based on applied category theory


NameDescriptionDevelopment Status
Catlab.jlA framework for applied category theory in the Julia languageCatlab.jl on GitHub
The package is nearing it's v1.0.0 release
AlgebraicPetri.jlBuild petri net agent based models compositionallyAlgebraicPetri.jl on GitHub
Functionality is mostly built-out, but the API may change substantially
AlgebraicRelations.jlBuild SQL categoricallyAlgebraicRelations.jl on GitHub
Functionality is mostly built-out, but the API may change substantially.
AlgebraicDynamics.jlBuild dynamical systems compositionallyAlgebraicDynamics.jl on GitHub
Functionality is mostly built-out, but the API may change substantially.
CombinatorialSpaces.jlSimplicial sets and other combinatorial models of geometric spacesCombinatorialSpaces.jl on GitHub
Functionality is mostly built-out, but the API may change substantially.

More info on these packages can be found below.


To learn more about applied category theory, try these introductory texts:

  • An invitation to applied category: Seven sketches in compositionality by Fong & Spivak (arXiv)

  • Category theory for the sciences by Spivak (arXiv)

The following resources introduce specific topics in, or perspectives on, applied category theory:

  • "Physics, topology, logic and computation: A Rosetta Stone" by Baez & Stay (arXiv), an interdisciplinary introduction to monoidal categories

  • "Categories for the practising physicist" by Coecke & Paquette (arXiv), another introduction to monoidal categories, emphasizing quantum physics and relations

  • Category theory for programmers blog by Milewski, a programming languages viewpoint on category theory with examples from C++ and Haskell, also available in PDF

  • Graphical linear algebra blog by Sobocinski, on the string diagrammatic approach to linear algebra

If your mathematical background includes basic abstract algebra, you might also try one of these more mathematical introductions to category theory:



  • A Categorical Representation Language and Computational System for Knowledge-Based Planning, 2023. Angeline Aguinaldo, Evan Patterson, James Fairbanks, William Regli, and Jaime Ruiz. 2023 AAAI Fall Symposium on Unifying Representations for Robot Application Development. arXiv, Slides

  • Structured and Decorated Cospans from the Viewpoint of Double Category Theory, 2023. Evan Patterson. Applied Category Theory 2023. arXiv, DOI

  • A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks, 2023. Rebekah Aduddell, James Fairbanks, Amit Kumar, Pablo S. Ocal, Evan Patterson, Brandon T. Shapiro. arXiv

  • Computational category-theoretic rewriting, 2023. Kristopher Brown, Evan Patterson, Tyler Hanks, James Fairbanks. Journal of Logical and Algebraic Methods in Programming. arXiv, DOI

  • Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves, 2023. Ernst Althaus, Benjamin Merlin Bumpus, James Fairbanks, Daniel Rosiak. arXiv


  • Compositional exploration of combinatorial scientific models, 2022. Kristopher Brown, Tyler Hanks, James Fairbanks. Applied Category Theory 2022. arXiv, Slides, Video

  • A diagrammatic view of differential equations in physics, 2022. Evan Patterson, Andrew Baas, Timothy Hosgood and James Fairbanks. Mathematics in Engineering. arXiv, DOI

  • Compositional modeling with stock and flow diagrams, 2022. John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel Osgood, Evan Patterson. Applied Category Theory 2022. arXiv, Slides, Video

  • An algebraic framework for rapid epidemic modeling, 2022. Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson, James Fairbanks. Accepted at Proceedings of the Royal Society A. arXiv

  • Computational category-theoretic rewriting, 2022. Kristopher Brown, Evan Patterson, Tyler Hanks, James Fairbanks. International Conference on Graph Transformation (ICGT 2022). arXiv, DOI

  • Categorical data structures for technical computing, 2022. Evan Patterson, Owen Lynch, James Fairbanks. Compositionality. arXiv, DOI


  • Operadic Modeling of Dynamical Systems: Mathematics and Computation, 2021. Sophie Libkind, Andrew Baas, Evan Patterson, James Fairbanks. Applied Category Theory 2021. arXiv, DOI


  • Wiring diagrams as normal forms for computing in symmetric monoidal categories, 2020. Evan Patterson, David Spivak, Dmitry Vagner. Applied Category Theory 2020. arXiv, Video

  • Compositional scientific computing with Catlab and SemanticModels, 2020. Micah Halter, Evan Patterson, Andrew Baas, James Fairbanks. Applied Category Theory 2020. arXiv



  • Abstraction and Composition in Modeling and Simulation, Luke Morris, Andrew Baas, Jesus Arias, Maia Gaitlin, James Fairbanks, SIAM Conference on Computational Science and Engineering, March 2023. Slides, Abstract, Schedule

  • Computational Category Theory in Applied Mathematics, Owen Lynch and James Fairbanks, Joint Mathematics Meetings, January 2023. Slides

  • Characterizing Compositionality of LQR from the Categorical Perspective Baike She, Tyler Hanks, James Fairbanks, Matthew Hale. Presented at IEEE Conference on Decision and Control., December, 2023. arXiv

  • Categorical Chemistry, Wilmer Leal, University of Pamplona Deparment of Mathematics, July 24, 2023.


  • AlgebraicJulia: a compositional approach to technical computing, Evan Patterson, NIST Workshop on Compositional Structures for Systems Engineering and Design, November 2022. Slides

  • Diagrammatic differential equations: Formal categorical framework and applications to multiphysics simulation (on arXiv:2204.01843), Timothy Hosgood, Applied Category Theory 2022, non-proceedings talk, July 2022. Slides, Video

  • Typed and stratified models with slice categories (on arXiv:2203.16345), Sophie Libkind, Applied Category Theory 2022, demo track, July 2022. Slides, Video

  • Individual.jl: Rewriting individual-based models for epidemiology using graph rewriting (on Individual.jl), Sean Wu, Applied Category Theory 2022, demo track, July 2022. Video

  • Compile-time programming with CompTime.jl, Owen Lynch, JuliaCon 2022, July 2022. Abstract, Video

  • Declarative data transformation via graph transformation, Kris Brown, JuliaCon 2022, July 2022. Abstract, Video

  • Principles and pitfalls of designing software for applied category theory, Evan Patterson, Topos Institute Berkeley Seminar, February 2022. Slides, Video


  • Categories of diagrams in data migration and computational physics, Evan Patterson, Topos Institute Colloquium, December 2021. Slides, Video

  • Shaped data with acsets, Owen Lynch, JuliaCon 2021, July 2021. Abstract, Video

  • Computational categorical algebra with Catlab, James Fairbanks, Graph Transformation Theory and Applications (GReTA) Seminar, May 2021. Video

  • C-sets for data analysis: graphs, relational data, and conjunctive queries, Evan Patterson, Category Theory and Applications Meetup, January 2021. Notebook, Video


  • (Co)relational computing in Catlab: The operad of UWDs and its algebras, Evan Patterson, MIT Categories Seminar, December 2020. Slides, Video

  • Implementing open dynamical systems in Catlab, Sophie Libkind, UNAM Categories Seminar, November 2020. Video

  • Compositional epidemiological modeling using structured cospans, Micah Halter and Evan Pattersion, UC Riverside Categories Seminar, November 2020. Slides, Video

  • AlgebraicJulia: Applied category theory in Julia, James Fairbanks, JuliaCon 2020, July 2020. Slides, Video

  • Realizing applied category theory in Julia, Evan Patterson, MIT Categories Seminar, January 2020. Video


Help by contributing code, asking questions, or reporting issues.

First off, thank you for your interest in AlgebraicJulia, no matter how you participate in the community.

The packages in AlgebraicJulia are open-source and liberally licensed to allow wide private and commercial usage of the packages, like the base Julia language and many other packages in the ecosystem.

Being open source, you are free to modify, use, or change your copy of the code - but if you make enhancements please consider opening a pull request (basic walkthrough here).

If you find issues, please open an issue on the relevant package's repository and we will try and address it as soon as possible.

If you have other ideas or questions, feel free to also open an issue, or discuss on the community Zulip or Slack #catlab channel. We welcome all Category Theory and related disciplines!