Dynamic categories, dynamic operads: From deep learning to prediction markets

Natural organized systems adapt to internal and external pressures and this seems to happens all the way down. 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 $\mathbf{Org}$ of dynamic organizations, we provide definitions of $\mathbf{Org}$-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.