Homotopy Wave Function in Algebraic Geometry

TL;DR

The work proposes a higher-homotopical generalization of the wave function by modeling natural phenomena as objects in the Segal topos of derived stacks $\mathcal{X} = \text{dSt}(k)$ and examining the comma topos $\mathcal{X}_{F/}$ to capture dynamical behavior. The main method constructs $\mathcal{X}$ via localization of simplicial $k$-algebras to derived affine stacks, then uses the homotopy shape $H_{\mathcal{X}_{F/}}$ to extract a higher, probabilistic wave-function-like descriptor of $F$, with the classical quantum wave function recovered in simple cases. Key contributions include the explicit formation of $\mathcal{X}_{F/}$ as a dynamical and homotopical refinement of $F$ and the definition of $H_{\mathcal{X}_{F/}}(Y)$ as a higher-wave function that remains meaningful under smooth deformations and t-completeness. The framework is generalized to arbitrary base model categories, enabling a multi-law or multi-context generalization via $\mathcal{X}_{\varphi/}$ and its homotopy shape, which broadens the scope for modeling complex physical and geometric phenomena.

Abstract

We propose the homotopy shape of the Segal topos of derived stacks over simplicial k-algebras as the higher homotopical generalization of the concept of wave function in Quantum Mechanics