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[2602.17917] Interactions that reshape the interfaces of the interacting parties
Summary
This paper introduces polynomial trees to model dynamic systems where interactions reshape interfaces, enhancing understanding of state-driven patterns in category theory and machine learning.
Why It Matters
The research provides a novel framework for understanding how systems evolve through interactions, which is crucial for advancements in machine learning and dynamic modeling. By integrating polynomial functors with dynamic organizations, it opens new avenues for designing adaptive systems, particularly in AI applications.
Key Takeaways
- Polynomial functors can model systems with dynamic interfaces.
- The concept of polynomial trees allows for coinductive modeling of interactions.
- This research generalizes existing frameworks in category theory to include evolving interfaces.
- Applications in generative adversarial networks illustrate practical implications.
- The study enhances the understanding of state-driven interaction patterns.
Mathematics > Category Theory arXiv:2602.17917 (math) [Submitted on 20 Feb 2026] Title:Interactions that reshape the interfaces of the interacting parties Authors:David I. Spivak View a PDF of the paper titled Interactions that reshape the interfaces of the interacting parties, by David I. Spivak View PDF HTML (experimental) Abstract:Polynomial functors model systems with interfaces: each polynomial specifies the outputs a system can produce and, for each output, the inputs it accepts. The bicategory $\mathbb{O}\mathbf{rg}$ of dynamic organizations \cite{spivak2021learners} gives a notion of state-driven interaction patterns that evolves over time, but each system's interface remains fixed throughout the interaction. Yet in many systems, the outputs sent and inputs received can reshape the interface itself: a cell differentiating in response to chemical signals gains or loses receptors; a sensor damaged by its input loses a channel; a neural network may grow its output resolution during training. Here we introduce *polynomial trees*, elements of the terminal $(u\triangleleft u)$-coalgebra where $u$ is the polynomial associated to a universe of sets, to model such systems: a polynomial tree is a coinductive tree whose nodes carry polynomials, and in which each round of interaction -- an output chosen and an input received -- determines a child tree, hence the next interface. We construct a monoidal closed category $\mathbf{PolyTr}$ of polynomial trees, with coinductively-de...
