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Organizing Physics with Open Energy-Driven Systems
Matteo Capucci, Owen Lynch, David I. Spivak
https://arxiv.org/abs/2404.16140 https://

Organizing Physics with Open Energy-Driven Systems
Organizing physics has been a long-standing preoccupation of applied category theory, going back at least to Lawvere. We contribute to this research thread by noticing that Hamiltonian mechanics and gradient descent depend crucially on a consistent choice of transformation -- which we call a reaction structure -- from the cotangent bundle to the tangent bundle. We then construct a compositional theory of reaction structures. Reaction-based systems offer a different perspective on composition in…

Pattern runs on matter: The free monad monad as a module over the cofree comonad comonad
Sophie Libkind, David I. Spivak
https://arxiv.org/abs/2404.16321 h…

Pattern runs on matter: The free monad monad as a module over the cofree comonad comonad
Interviews run on people, programs run on operating systems, voting schemes run on voters, games run on players. Each of these is an example of the abstraction pattern runs on matter. Pattern determines the decision tree that governs how a situation can unfold, while matter responds with decisions at each juncture. In this article, we will give a straightforward and concrete construction of the free monad monad for $(\mathbf{Poly}, \mathbin{\triangleleft}, \mathcal{y})$, the category of polyn…