Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interactions that reshape the interfaces of the interacting parties

David I. Spivak

TL;DR

A locally fully faithful functor is provided via constant trees, those for which the interfaces do not change through time, of progressive generative adversarial networks, where gradient feedback determines when the image-generation interface grows to a higher resolution.

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-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.

Interactions that reshape the interfaces of the interacting parties

TL;DR

A locally fully faithful functor is provided via constant trees, those for which the interfaces do not change through time, of progressive generative adversarial networks, where gradient feedback determines when the image-generation interface grows to a higher resolution.

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 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 of polynomial trees, with coinductively-defined morphisms, tensor product, and internal hom. We then build a bicategory generalizing , whose hom-categories parametrize morphisms by state sets with coinductive action-and-update data. We provide a locally fully faithful functor 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.
Paper Structure (17 sections, 11 theorems, 34 equations)

This paper contains 17 sections, 11 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Preliminaries
  4. Polynomial functors and their maps
  5. Monoidal structures on $\mathbf{Poly}$
  6. Dirichlet product and closure.
  7. Substitution product.
  8. The cofree comonad
  9. Universe polynomials
  10. The category $\mathbf{PolyTr}_\mathbb{U}$ of polynomial trees
  11. Polynomial trees
  12. The category of polynomial trees
  13. Monoidal structure
  14. Monoidal closure
  15. Relationship to $\mathbf{Poly}$ and $\mathbb{O}\mathbf{rg}$; the bicategory $\mathbb{O}\mathbf{rgTr}$
  16. ...and 2 more sections

Key Result

Proposition 1

The forgetful functor $U\colon\mathbf{Cmd}(\mathbf{Poly})\to\mathbf{Poly}$ has a right adjoint $\mathfrak{c}\colon\mathbf{Poly}\to\mathbf{Cmd}(\mathbf{Poly})$, \begin{tikzcd}[ampersand replacement=\&, column sep=30pt] \Cmd(\poly)\ar[r, shift left=5pt, "U"] \ar[r, phantom, "\scriptstyle\Rightarro with projection maps $\pi^{(i)}\colon p^{(i+1)}\to p^{(i)}$, where $\pi^{(0)}$ is the terminal map

Theorems & Definitions (38)

  • Remark 1
  • Proposition 1: libkind2024pattern
  • Definition 3.1
  • Example 1: Communication protocols
  • Example 2: Cell differentiation
  • Example 3: Chess
  • Definition 3.2
  • Example 4: Protocol refinement
  • Example 5: Maps from and to $\overline{\mathtt{{\mathcal{y}}}}$
  • Proposition 2
  • ...and 28 more