Dynamic Operads, Dynamic Categories: From Deep Learning to Prediction Markets

TL;DR

This paper introduces a category-theoretic framework for modeling adaptive systems like deep learning and prediction markets as dynamic categorical structures, providing a formal mathematical foundation for their evolution and organization.

Contribution

It develops the monoidal double category Org of dynamic organizations and defines dynamic categorical structures such as dynamic operads and monoidal categories, linking philosophical ideas to formal category theory.

Findings

Prediction markets modeled as dynamic operads

Deep learning represented as dynamic monoidal categories

Formal category-theoretic framework for adaptive systems

Abstract

Natural organized systems adapt to internal and external pressures and this happens at all levels of the abstraction hierarchy. Wanting to think clearly about this idea motivates our paper, and so the idea is elaborated extensively in the introduction, which should be broadly accessible to a philosophically-interested audience. In the remaining sections, we turn to more compressed category theory. We define the monoidal double category Org of dynamic organizations, we provide definitions of Org-enriched, or dynamic, categorical structures -- e.g. dynamic categories, operads, and monoidal categories -- and we show how they instantiate the motivating philosophical ideas. We give two examples of dynamic categorical structures: prediction markets as a dynamic operad and deep learning as a dynamic monoidal category.