Interactions that reshape the interfaces of the interacting parties
David I. Spivak

TL;DR
This paper introduces polynomial trees to model systems with dynamic interfaces that evolve during interactions, extending existing frameworks to capture systems like neural networks and chemical cells that change their input/output capabilities over time.
Contribution
It develops a new categorical framework using polynomial trees and bicategories to model systems with interfaces that reshape during interactions, generalizing previous fixed-interface models.
Findings
Constructed a monoidal closed category of polynomial trees.
Built a bicategory of systems with evolving interfaces.
Illustrated applications to generative adversarial networks.
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 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 -coalgebra where is the polynomial associated to a universe of sets, to model such systems: a polynomial tree is a coinductive tree…
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Taxonomy
TopicsNeural Networks and Reservoir Computing · Model Reduction and Neural Networks · Topological and Geometric Data Analysis
