John Baez

Geometry of Complex Webs 2026, November 1-6, 2026

Applied Category Theory and the Mathematics of Networks

Engineers and scientists study and use networks of many kinds. Applied category theory provides a unified approach to networks with many advantages. After overviewing the field, we define monoidal categories and monoidal functors, and then illustrate their use for working with "stock and flow diagrams". These are widely used for modeling in epidemiology.   Some modelers regard these diagrams as an informal step toward a mathematically rigorous formulation of a model in terms of ordinary differential equations.  However, these diagrams have a precise syntax, which can be explicated using category theory.  I will explain some software that takes advantage of these ideas. Basing this software on category theory has many advantages, but I will explain three: model composition, functorial semantics, and model stratification. 

You can see my slides here:

• Part 1: open networks and stock-flow diagrams.
• Part 2: monoidal categories and monoidal functors.
• Part 3: software for working with the monoidal category of open stock-flow diagrams.

For more, read these:

• Xiaoyan Li, Evan Patterson, Patricia L. Mabry and Nathaniel D. Osgood, Compositional system dynamics: the higher mathematics underlying system dynamics diagrams and practice.
• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams, in Mathematics of Public Health: Mathematical Modelling from the Next Generation, eds. Jummy David and Jianhong Wu, Springer, 2023, pp. 175-207.
• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams, Proceedings Fifth International Conference on Applied Category Theory, EPTCS 380 (2022), 77-96.
• Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James Fairbanks, An algebraic framework for structured epidemic modeling, Phil. Trans. R. Soc. A. 380 (2022), 20210309.
• Evan Patterson and Micah Halter, Compositional epidemiological modeling using structured cospans.

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