Interactions that reshape the interfaces of the interacting parties
This work addresses a foundational gap in mathematical modeling for dynamic systems in fields like biology and machine learning, where interfaces evolve over time, though it is incremental by extending existing polynomial functor frameworks.
The paper tackles the problem of modeling systems whose interfaces can change during interactions, such as cells differentiating or neural networks growing resolution, by introducing polynomial trees as a coinductive structure where each interaction round determines the next interface. They construct a monoidal closed category and bicategory for these systems, generalizing previous fixed-interface models, and illustrate it with progressive generative adversarial networks.
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-defined morphisms, tensor product, and internal hom. We then build a bicategory $\mathbb{O}\mathbf{rgTr}$ generalizing $\mathbb{O}\mathbf{rg}$, whose hom-categories parametrize morphisms by state sets with coinductive action-and-update data. We provide a locally fully faithful functor $\mathbb{O}\mathbf{rg}\to\mathbb{O}\mathbf{rgTr}$ via constant trees, those for which the interfaces do not change through time. We illustrate the generalization by suggesting a notion of progressive generative adversarial networks, where gradient feedback determines when the image-generation interface grows to a higher resolution.