[论文解读] Interactions that reshape the interfaces of the interacting parties

David I. Spivak|arXiv (Cornell University)|Feb 20, 2026

Neural Networks and Reservoir Computing被引用 0

一句话总结

论文引入多项式树来建模在交互过程中接口会演化的系统，并构建单合成闭合范畴 PolyTr 与双范畴 OrgTr，以捕捉共同演化的接线与接口，及其在渐进生成模型中的应用。

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 $(u riangleleft u)$-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 $\mathbb{O}\mathbf{rg} o\mathbb{O}\mathbf{rgTr}$ 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.

研究动机与目标

• 为系统的接口（不仅仅是接线）在系统流动中改变提供动机与建模
• 扩展现有的动态组织框架以适应演化接口
• 开发一个数学生态（PolyTr 与 OrgTr），支持共诱导态射、张量积与内部同态，以实现演化接口的建模

提出的方法

• 将多项式树定义为终端（u ▷ u）-煤炭代数的元素，其中 u 是宇宙多项式
• 构建带有共诱导定义态射的范畴 PolyTr(U)，具备张量积和单合成闭结构
• 通过常数树将具有固定接口的多项式嵌入 PolyTr(U)
• 将 Org 推广为 OrgTr，作为双范畴，其态射参数化为具有共诱导状态更新数据的状态集合
• 给出从 Org 到 OrgTr 的保真函子，识别常数树情形下的固定接口模型并嵌入演化接口框架

实验结果

研究问题

• RQ1 接口本身在交互中如何改变，以及如何在范畴论中形式化？
• RQ2 哪些范畴结构（单合成闭、共诱导态射）同时支持演化接口和接线？
• RQ3 如何将固定接口模型与演化接口框架联系起来？
• RQ4 有哪些具体应用能说明演化接口动力学，如渐进生成过程？

主要发现

• 构建了带共诱导定义态射的多项式树的单合成闭范畴 PolyTr(U)
• 双范畴 OrgTr 通过将态射参数化为共诱导状态更新数据来一般化 Org
• 存在一个从 Org 到 OrgTr 的局部全忠实函子，借助常数树将固定接口模型嵌入演化接口框架
• 该框架支持富化，进而形成动态范畴、操作子以及张量范畴等，其中接线与接口共同演化
• 一个来自渐进深度学习的例子说明了 OrgTr 构造在实际情境中的应用